lesson masters a -...

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Vocabulary

1. What is a pixel?

2. How is the resolution of a computer screen measured?

Skills Objective AIn 3–6, draw the line or lines described.

3. an oblique discrete line 4. a vertical discrete line

5. a vertical discrete line intersecting 6. two oblique discrete lines thata horizontal discrete line at a point cross but have no point in common

Properties Objective FIn 7 and 8, true or false.

7. In discrete geometry, a point has no size.

8. In discrete geometry, two lines mayintersect at two distinct points.

Uses Objective J

9. What is represented by the discreteline in the road-atlas legend atthe right?

Questions on SPUR ObjectivesSee pages 58–61 for objectives.1-1

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Toll Limited-Access HighwayPrincipal HighwayOther RoadUnpaved RoadScenic Route

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Properties Objective F

1. True or false. A point as a location has thickness.

2. Describe a property that holds for lines in syntheticgeometry but not for lines in discrete geometry.

3. Give a possible coordinate forpoint B on the number line atthe right.

Uses Objective I

4. At the Mt. Washington, NH, weather station the greatestwind speed on record is 231 mph, while the average windspeed is 35.3 mph. The greatest wind speed is how much more than the average?

5. A tailor had her customers stand on a 9 0-high platform to mark hems with a yardstick. If she marked a hem at 31 0from the floor, how far would the hem actually beoff the ground?

6. According to the 1995 Information Please Almanac, the record high temperature in Antarctica was 15˚C in 1974,and the record low temperature was -89˚ C in 1983. Howmuch colder was the record low than the record high?

Representations Objective K

In 7–8, use the number line atthe right.

7. Find the distance from a. A to C. b. C to B.

8. If F is on this number line, and AF 5 12,give the two possible coordinates of F.

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0 3

A CB

100 101

A CB

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1. True or false. In coordinate geometry, lines are dense.

2. Name two number lines in theCartesian plane.

Representations Objective LIn 3–6, give two points on the line with thegiven equation. Then graph the line.

3. y 5 -4x 1 6 4. x 1 2y 5 8

5. y 5 -3 6. x 5 4

In 7–9, use the equation to classify the line asvertical, horizontal, or oblique.

7. y 5 -462 8. 7x 1 y 5 1 9. x 5 1259

x yx y

x yx y

Questions on SPUR ObjectivesSee pages 58–61 for objectives.

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Skills Objective B

1. Consider the network at the right.

a. How many odd nodes are there?

b. How many even nodes are there?

c. Is the network traversable? If it is, give a path.

2. Draw a network consisting of five lines and three nodes.


3. In graph theory, there may be four distinct lines betweentwo points.

4. In graph theory, a point has size and shape.

5. Multiple choice. A line is dense in which geometry?

(a) discrete geometry

(b) plane coordinate geometry

(c) graph theory


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Uses Objective J

6. Use the street plan at the right.

a. Model the street plan as a network of lines and nodes.

b. Would a mail carrier be able to plan a route to visit all the houses on the streets without retracing his steps? Why or why not?

7. At the right is a diagram of San FranciscoBay and the counties bordering it. The bayis crossed by 7 toll bridges.

a. Represent the bridges and land massesas a network of lines and nodes.

b. Is the network traversable?

c. In 1989, the Bay Bridge, which connects San FranciscoCounty with Alameda County, was closed after an earthquake.Without the use of the Bay Bridge, is the network traversable?Justify your answer.

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Sonoma Co.

San Francisco Co.

Solano Co.Marin Co.

Contra Costa Co.

San Mateo Co.

Alameda Co.

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Skills Objective C

In 1–2, tell whether the drawing is a perspective drawing.

1. 2.

3. Change the drawing to a perspective drawing.

In 4–6, locate the vanishing point of the drawing.

4. 5. 6.


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MILK

milk carton

highway buildingtable tennis

football field

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Properties Objective EIn 1 and 2, tell how many dimensions the object has.Ignore small thicknesses.

1. a shoebox 2. a paper clip

Properties Objective GIn 3–6, in what type of geometry could the triangle appear?

3. 4.

5. 6.

In 7–9, true or false. If false, rewrite to make it true.

7. Space is an undefined term in geometry.

8. A line looks the same in any geometry.

9. Coordinate geometry is the study offigures in three dimensions.

In 10–12, multiple choice. Choose the geometry in which the statement is true.

10. A line is dense.

(a) graph theory (b) synthetic geometry (c) discrete geometry

11. More than one distinct line may contain two given points.

(a) graph theory (b) plane coordinate geometry (c) synthetic geometry

12. Points have a definite shape.

(a) synthetic geometry (b) discrete geometry (c) plane coordinate geometry


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Properties Objective G

1. Which two geometries are Euclidean?

2. What is the difference between a postulate and a theorem?

In 3–6, tell which assumption from the Point-Line-Plane Postulate permits the procedure described.

Unique Line Assumption Number Line Assumption

Dimension Assumption

3. Locating point P not on line m.

4. Graphing the line with equation y 5 2x byconnecting the ordered pairs (1, 2) and (2, 4).

5. Locating point P not on plane m.

6. Measuring the length of a segment by holdinga ruler to it.

In 7–12, tell if the statement is true or false in Euclidean Geometry.

7. A point can be represented by an ordered pair of real numbers.

8. Lines have no thickness.

9. Any line may be coordinatized using the real numbers.

10. Two different lines cannot intersect at two different points.

11. Lines are not dense.

12. Lines are parallel if they have no points in common or ifthey are the same line.


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Skills Objective D

In 1–4, use←→LP at the right.

1. Name↔LP in two other ways. 2. Name

→ON in two other ways.

3. True or false. MN 5 MN

4. Name the ray opposite to →MN.

5. Draw →TS with M between T and S.

6. Draw AB with S between A and B, and M between S and B.

Properties Objective HIn 7 and 8, use the figure at the right. A isbetween M and P.

7. If MA 5 17.3 and AP 5 9.7, what is MP?

8. If AM 5 36 and PM 5 92, what is AP?

9. On a number line O is between X and Y. If X hascoordinate -18 and Y has coordinate -2, give therange of coordinates for O.

10. On a number line, P is between L and M. If L hascoordinate -20, M has coordinate 3, and LP 5 5,what is the coordinate of P?

In 11 and 12, use the number lineat the right. Write an inequality todescribe the coordinates of points on

11.→BD. 12. AC.


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-40 -20 20 400

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Skills Objective AIn 1–6, tell whether the set is convex or nonconvex.

1. 2. 3.

4. 5. 6.

In 7 and 8, draw the figure.

7. a convex 10-sided region 8. a nonconvex 6-sided region

9. Draw a segment to show that the set atthe right is not convex.

10. Imagine that a friend cannot understand why Figure 1 below is aa convex set, but Figure 2 is not convex. Write an explanation to help your friend with the problem.

Fig. 1 Fig. 2


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Skills Objective CIn 1–3, let r 5 the cake falls; s 5 you stamp your foot; and t 5 the floor shakes. Write the conditional in words.

1. t ⇒ r

2. s ⇒ t

3. s ⇒ r

4. Write in symbols: If a, then b.

Properties Objective GIn 5 and 6, underline the antecedent once and the consequent twice.

5. If wishes were horses, then beggars would ride.

6. Make yourself a sandwich if you’re hungry.

In 7 and 8, rewrite as a conditional.

7. All 21-year-olds are eligible to vote.

8. Every sign must be written in English and in Spanish.

Properties Objective H

9. Consider the BASIC 10 INPUT Qprogram at the right. 20 IF Q >= 15 THEN PRINT “GOODBYE”

30 END

a. Give a value for Q which will cause the computerto print “GOODBYE”.

b. Give a value for Q which will cause the computer to printnothing. Explain your answer using a conditional statement.

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10. True or false. An instance of “if w, then z” is a situationin which w is true and z is false.

In 11 and 12, a conjecture is given. Determinewhether each example is

(i) an instance of the conjecture.

(ii) a counterexample to the conjecture.

(iii) neither an instance nor a counterexample to the conjecture.

11. If t ≥ 40, then t ≥ 41.

a. t 5 45 b. t 5 39 c. t 5 40.3

12. If a figure has four sides, then its interior is a convex set.

a. b. c.

Uses Objective K

13. An ad said, “If you buy a refrigerator before Friday, you’llreceive a $100 rebate.” Danielle bought a refrigerator onThursday of the same week. What will happen?

14. If the service at a restaurant is very good, then Lonnie willleave a 20% tip. Today, Lonnie thought the service was verygood. What will happen?

15. Rewrite as a conditional: It is always cloudy when it rains.

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Skills Objective C

1. Write the converse of c ⇒ d.

Properties Objective EIn 2 and 3, true or false.

2. The converse of a true statement is always true.

3. The converse of a false statement is always false.

In 4 and 5, a conditional is given. a. Is the conditionaltrue? If not, give a counterexample. b. Write its converse.c. Is the converse true? If not, give a counterexample.

4. If the light is red, the traffic stops.

a.

b.

c.

5. If m > 0, then m ≥ 2.

a.

b.

c.

6. Use the consequent You can vote in the U.S. presidential election.Write a conditional which is true but whose converse is not true.

Uses Objective K

7. An ad said: “If you use our shampoo, your hair will be clean andshiny.” Jessica read the ad and wanted clean and shiny hair. A month later her hair was very clean and quite shiny. Did Jessica usethe shampoo? Write an explanation for your answer in which you use the converse of the conditional in the ad.


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Skills Objective C

1. Let p 5 water freezes and q 5 the temperature is lessthan or equal to 08 C. Write p ⇔ q in words.

Skills Objective DIn 2 and 3, use the number line atthe right.

2. Find the coordinate of each point.

a. the midpoint of AB

b. the midpoint of BC

c. the midpoint of AC

3. If A is the midpoint of CD, what is the coordinate of D?

4. M is the midpoint of GH,GM 5 2(x 1 5), andMH 5 10x 2 13.Find GH.

5. X is equidistant from Z and W,XW 5 9 2 2t, and ZW 5 6t 2 9.Find ZX.


6. Consider this definition of tangent: A tangent to a circle is a linewhich intersects the circle in exactly one point. Name two undefinedgeometric terms used in this definition.

Z WX

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-300 -200 -100 0 100 200 300

B C


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7. Consider the definition for a chord of a circle: A segment is achord of a circle ⇔ its endpoints lie on the circle.

a. Write this definition as two conditionals.

b. Which conditional goes in the direction characteristics ⇒ term?

In 8–10, tell which property of a good definition isviolated by the “bad” definition.

8. A polygon is a figure with sides and angles.

9. A line segment is a straight path between two points.

10. A line segment consists of two points and all the pointsbetween them, and it is always straight and can be measuredto find its length.

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Vocabulary

1. What is the meaning of each symbol?

a. > ________________________ b. < ________________________

c. { } ________________________

Properties Objective I

2. Let M 5 {100, 200, 300, 400, 500} and N 5 {0, 1, 2, 3, . . ., 200}.Identify the set.

a. M > N _____________________ b. M < N _____________________

3. Let R 5 set of real numbers s with s ≥ 15 andP 5 set of real numbers s with s ≤ 20.

a. Draw R and P on thenumber line at the right.

b. Write an inequality for R > P.

c. Describe R < P.

4. Use the figure at the right. Give

a. the segment(s) of DFER > DREU.

b. the segment(s) of DFER < DREU.

.c. the segment(s) of quadrilateral FIGU > DFRU.

d. the segment(s) of quadrilateral FIGU > DERU.

e. the point(s) of IG > ER.

f. the segment(s) of pentagon FIGUR < DFRE.

5. Let G 5 the set of all states east of the Mississippi River and H 5 the set of all states west of the Mississippi River. Describe each set.

a. G < H _______________________ b. G > H _____________________

6. Draw→PT and

→RT 7. Draw

←→MN and AB

so that→PT >

→RT 5 T. so that

←→MN > AB 5 AB.

M B A NP RT

I

U

R

F

G

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-10 -5 0 5 10 15 20 25P

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Skills Objectives A and B

1. Draw a nonconvex decagon. 2. Draw a convex pentagon.

3. Use the definition of polygon to explain why the figureat the right is not a polygon.

4. Draw a figure that is the union of four segments but is not a quadrilateral.

Uses Objective L

5. Some tables for playing games are shaped like the polygonat the right.

a. Name the polygon.

b. Give a reason for its shape.

6. The playing area at Oriole Park at Camden Yards inBaltimore, MD, is diagrammed at the right.

a. What name is given to the large polygon?

b. The small figure is not a polygon. Why?

Representations Objective N7. Draw a hierarchy relating the following

terms: figure, polygon, triangle, quadrilateral, isosceles triangle, nonagon.

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Properties Objective JIn 1–5, can the numbers be lengths of the sides ofa triangle? Justify your answer.

1. 10, 15, 25

2. 4, 4, 4

3. 4, 15, 11

4. , ,

5. 1, 10, 10

6. Stu drew and labeled the triangle below. Why didhis teacher give him no credit for his drawing?

7. Two sides of a triangle measure 92 meters and54 meters. How long can the other side be?

8. In DISO at the right, if IS 5 30 how long can SO be?

Uses Objective M

9. It is a 35-minute drive from Kyoko’s house to Anne’s house and a 25-minute drive from Anne’s house to Ben’s house.Using just this information, tell how much time it wouldtake to drive from Kyoko’s house to Ben’s house.

10. It is 44 miles from Lancaster, Pennsylvania, to Harrisburg,Pennsylvania, and 78 miles from Harrisburg to Reading,Pennsylvania. Using just this information, tell how farit is from Reading to Lancaster.

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Properties Objective HIn 1–3, true or false.

1. A conjecture is always true.

2. One counterexample will show that a conjecture is false.

3. People make conjectures based on patterns they observe.

In 4–7, a conjecture is given. Determine whethereach example is an instance of the conjecture, acounterexample to the conjecture, or neither aninstance nor a counterexample of the conjecture.

4. If x is a positive number, x 2 ≥ 1.

a. x 5 5 b. x 5 -2 c. x 5 .5

5. If x ≤ 0, then x ≤ -1.

a. x 5 10 b. x 5 - c. x 5 -3

6. If the area of rectangle RECT is 100, then the perimeter of RECT is 40.

a. b. c.

7. If XT 5 TY, then T is the midpoint of XY.

a. b. c.

X

Y

T5.0

5.0

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5.0

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4.9

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Skills Objective A

1. Draw and label an ∠ CDB 2. Draw and label an ∠ XTL with

with measure 124. measure 86 and its bisector→TA.

In 3–7, use the figure at the right.

3. Give two other names for ∠ APT.

4. Name two distinct straight angles.

5. True or false. ∠ LTP has measure 0.

6. m∠ APT 1 m∠ TPR 5 m∠ APR is an exampleof the ? Property.

7. If m∠ LPR 5 127, find each measure.

a. m∠ LPE b. m∠ TPR

c. m∠ TPA d. m∠ APL

In 8–10, true or false. Use the diagramat the right. You can assume from thediagram that

8. X, W, and Z are collinear.

9. →WY bisects ∠ QWZ.

10. Y is in the interior of ∠ QWZ.


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Skills Objective B

11. Suppose m∠ ATC 5 145, m∠ ATY 5 6b 1 10, and m∠ CTY 5 3b 1 9.

a. Find b.

b. Find m∠ ATY.

12. Suppose →IT bisects m∠ BIS,

m∠ BIT 5 12x 1 3, and m∠ TIS5 10x 1 10.

Find m∠ BIS.

13. Let m∠ POR 5 5t, m∠ ROL 5 3t 1 8, and m∠ POL 5 9t 2 1.

a. Write an equation to find t.

b. Find t and each measure.

t m∠ POR m∠ ROL m∠ POL

c. Does→OR bisect ∠ POL?

14. Suppose in the figure for Question 13,→OR bisects ∠ POL,

m∠ POR 5 5t, and m∠ POL 5 9t 1 10.

a. Write an equation involving t. b. Find t.


15. What property did you use to writethe equation in Question 13a?

16. What justification allows you to writethe equation in Question 14a?

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Skills Objective E

1. Rotate AB 658 about O. 2. Rotate DPQR -908 about M.

3. Draw the image of MNOP after it 4. B is the image of A after ahas been rotated -908 about O. rotation about point O. What

is the magnitude of the rotation?

In 5 and 6, true or false.

5. A rotation of 908 is equivalent to a rotation of -908.

6. A rotation of 1808 is equivalent to a rotation of -1808.

Skills Objective F

7. Use (O at the right, in which mXBC 5 33° andm∠ COD 5 90. Find each measure.

a. m∠ BOC

b. mXAB

c. mXBCD

d. mXBAC


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8. Use (P at the right, in which PT bisects ∠ SPO and m∠ SPD 5 50. Find each measure.

a. mXSDO

b. mXDT

c. mXTOD

Uses Objective I

9. A pendulum swings from point A topoint B and back. Measure to find themagnitude of the rotation

a. from B to A.

b. from A to B.

10. A circular cake is to be cut into 10 wedgesof the same shape and size.

a. What is the degree measureof the arc of each wedge?

b. If two people don’t wantcake and the cake is insteaddivided equally among the rest of the guests, what is the degree measure of the arc of one wedge?

11. The dial on a clothes dryer can beset from Hot to Air and to four settings in between. Arc measures from any position to the next are the same. If the dial is turned from 1 to 4, what is the magnitude of the rotation?

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Skills Objective AIn 1–3, use the diagram at the right.

1. Name two linear pairs.

2. Name a pair of vertical angles.

3. If m∠ OTB 5 45, find the measure of as many otherangles as possible.

4. a. If m∠ 8 5 112, then m∠ 9 5

and m∠ 10 5 .

b. If m∠ 8 5 10k, m∠ 11 5

and m∠ 10 5 .

Skills Objective B

5. Suppose m∠ AOB 5 10x 2 6 and m∠ DOC 5 14x 2 20.Find each measure.

a. m∠ DOC b. m∠ BOD

6. Suppose m∠ 1 5 11n 1 13 and m∠ 4 5 5n 2 9.

a. Find m∠ 3.

b. Explain how you found the answer in Part a.


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6. Two angles are complementary. The measure of thelarger is 10 more than 4 times the measure of thesmaller. Find the measure of each angle.

7. Students were asked to draw two non-adjacent supplementaryangles with the one angle 4 times as large as the other.

Art drew this diagram. Augie drew this diagram.

Neither student received full credit. Explain why and drawa correct answer.

Uses Objective I

8. Draw the path of a plane flying 758 9. A plane’s course is 608 east ofsouth of west. north. How far must it turn to

change its course to due east?

10. A mirror which is hinged to a wall is moved so thatit makes an 838 angle with the wall. What is themeasure of the other angle it makes with the wall?

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144° 36°

72° 18°

36°144°

75°EW

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EW

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Properties Objective GIn 1–5, multiple choice. Each statement illustrates one of theproperties listed at the right. Give the letter of the property.

1. If 1.7g2 5 22, then 22 5 1.7g2.

2. If 3x 1 5 5 -2, then 3x 5 -7.

3. m∠ ABC 5 m∠ CBA.

4. If GK 5 MN and MN 5 RQ,then GK 5 RQ.

5. If t 5 9, then t 5 27.

In 6 and 7, use the figure at the right.

6. If AB 1 BL 5 19 and BL 5 LE, whatcan you conclude about AB and LE by theSubstitution Property?

7. If AB 5 BL and LE 5 AB, what canyou conclude by the Transitive Property?

8. a. Write an equation relating ∠ PQR, ∠ PQS,and ∠ RQS.

b. Use the Equation to Inequality Property to write atrue statement involving ∠ SQR and ∠ PQR.

c. Is it possible to conclude that m∠ SQR < ∠ PQS?Give a reason for your answer.

9. Suppose m∠ T 5 90 2 x. Find m∠ T if

a. x 5 49. b. x 5 5a.

c. What property did you use in Parts a and b?

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(a) Reflexive Property of Equality

(b) Symmetric Property of Equality

(c) Transitive Property of Equality

(d) Equation to Inequality Property

(e) Substitution Property

(f) Addition Property of Equality

(g) Multiplication Property ofEquality

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Properties Objective HIn 1–3, multiple choice. Choose the correct justificationfor the given conclusion.

1. Given: m∠ POT 5 109.Conclusion: ∠ POT is obtuse.

(a) Angle Addition Postulate

(b) Vertical Angles Theorem

(c) definition of obtuse angle

2. Given: I is the midpoint of MN.Conclusion: MI 5 IN

(a) Additive Property for Segments

(b) definition of midpoint

(c) Equation to Inequality Property

3. Given: ∠ 4 and ∠ 3 are a linear pair.Conclusion: ∠ 3 and ∠ 4 are supplementary.

(a) definition of supplementary angles

(b) Linear Pair Theorem

(c) Angle Addition Postulate

In 4–6, write a justification for each conclusion.

4. Given: MA 5 AN.Conclusion: DMAN is isosceles.

5. Given ∠ POT and ∠ ROE are vertical angles.Conclusion: m∠ POT5 m∠ ROE.

6. Given: (P.Conclusion: T and L are equidistant from P.


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Skills Objective AIn 1–4, use the figure at the right inwhich m // <.

1. Name the transversal of m and ,.

2. Name two pairs of corresponding angles.

3. If m∠ 2 5 117, find the measure of each angle.

∠ 1 ∠ 3 ∠ 4 ∠ 5 ∠ 6

4. If m∠ 1 5 4e, find the measure of each angle.

∠ 2 ∠ 3 ∠ 4 ∠ 5 ∠ 6

5. Use the figure at the right in which p // q.If m∠ 1 5 5j 2 27 and m∠ 2 5 3j 1 5, findthe measure of each angle.

∠ 1 ∠ 2 ∠ 3

∠ 4 ∠ 5

Properties Objective HIn 6–8, in the figure at the right, < // m. Justifythe conclusion.

6. m∠ 2 5 m∠ 8.

7. m∠ 1 5 m∠ 3.

8. ∠ 3 and ∠ 6 are supplementary.


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56 3

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5 12

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In 9 and 10, name the theorem which justifies the conclusion.

9. Line r is parallel to line i. Line i is parallel to line t.Line t is parallel to line e. Therefore, line r is parallel to line e.

10. Line r has slope - and line t has slope - .Therefore, line r is parallel to line t.


11. Give the slope of the line passingthrough (-4, 1) and (-5, -3).

12. If c ≠ a, give the slope of the linethrough (a, b) and (c, d ).

In 13–15, give the slope of each line inthe figure at the right.

13.←→MP

14.←→PQ

15.←→MQ

16. Find the slope of each line.

a. y 5 - x 1 7 b. -5x 2 y 5 10

Representations Objective L

17. What is the slope of any line parallel toa line with slope -1.3?

18. What is the slope of any line parallel to←→PQ in Questions 13–15?

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x

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Q(7, 1)

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Skills Objective C

1. In the figure at the right,←→EF ⊥

←→FG

and m∠ EFG 5 3x 2 7.

Find x.

2. Given: s ⊥ r; m∠ 1 5 12w 2 6.

a. Find w.

b. If m∠ 2 5 14w 2 22,what is m∠ 2?

c. Is t ⊥ r? d. Is s // t?

Properties Objective HIn 3–6, use quadrilateral MNOPat the right.

Given: MP←→

// ←→QR,

←→QR ⊥

←→MN,

←→QR ⊥ PR

←→, and

←→NO //

←→QR.

Multiple choice. Choose the correctjustification for the conclusion.

3.←→MP //

←→NO.

4. ∠ RQN is a right angle.

5.←→MP ⊥

←→PR.

6.←→MN //

←→PO.


7. Find the slope of any line perpendicular to a line

a. with slope . b. with equation y 5 -8x 1 5.

c. with equation -2x 1 4y 5 9.

8. a. Find the slope of ←→PQ.

b. Give the slope of any line ⊥ to ←→PQ.

c. Draw the line ⊥ to ←→PQ through the origin.

45

(a) Two Perpendiculars Theorem

(b) Perpendicular to ParallelsTheorems

(c) definition of perpendicular line

(d) Transitivity of ParallelismTheorem


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Skills Objective DIn 1 and 2, construct the perpendicular bisectorof the given segment.

1. 2.

3. Use the figure at the right.

a. Draw the line through Q ⊥ to m.Label the intersection R.

b. Draw the line through Q // to m.

Label the line ←→QT.

c. How are ←→QT and

←→QR related?

Uses Objective J

4. The home at the right needs a gas pipeconnected to the gas main under the street.If the furnace is located in the southwestcorner of the home, draw the shortestpipe from the furnace to the gas main. Themain electrical line runs parallel to the gasmain. Draw the shortest line from theelectrical box to the main electrical line.

5. The lines drawn in Question 4 are

.

6. A park district built a foot path from theparking lot to the pond at the right. Theywish to add a parallel road for trucks fromthe parking lot to the end of the pond.Draw in the road for the trucks.

7. A path is to be made from the truck roadto the bait shop. Draw the shortest path.


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Q

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T

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furnaceelectricalbox

STREETGAS MAIN

MAIN ELECTRICAL

PARKING LOT

Bait Shop

Path

POND

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Skills Objective A

1. a. Draw rs(P).

b. Draw rs(T).


a. rp(A) 5 b. rs(C) 5

c. rt(D) 5 d. rs(E) 5

3. Draw the reflecting line 4. Can r,(T) 5 T'? Give a reason forp so that rp(M) 5 N. your answer.

Properties Objective EIn 5–8, true or false.

5. If rs(P) 5 P', then s is the perpendicularbisector of PP'.

6. If P is on ,, then r, (P) does not exist.

7. If r←→MN

(P) 5 P', then PP' bisects ←→MN.

8. A reflection is a transformation thatmaps a preimage onto an image.

Representations Objective KIn 9–12, give the coordinates of the image.

9. rx-axis(-7, 4) 10. ry-axis(2, 1)

11. rx-axis(-6, 0) 12. rx-axis(m, n)


A

P

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= P'

s

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sqp t

N

Mp

T

T'

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1. Draw r,(PQ). 2. Draw rm(TOMN). 3. Draw r←→AB

(G).

4. Draw the reflecting line such that figure II 5. Draw and label a figure for thisis the image of figure I over that line. condition r←→

AB(SAME) 5 PANT

Properties Objective E6. True or false. Reflections preserve angle measure,

distance, and orientation.

7. In the figure at the right, r←→OT

(MOT) 5 SOT.

a. If m∠ MOT 5 19, then m∠ MOS 5 .

b. m∠ MXO 5 .

c. If MS 5 10.10, then MX 5 5 .

d. If MT 5 120, then TS 5 .


8. a. Draw ry-axis(PQRS).

b. List the coordinates of the vertices ofrx-axis(PQRS).


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Q'P',

O

T

T'

M' m

MN

GG'A

B

I II

S ME

A

P

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T

M

S

X

R

x

y

2

-6-2

6P

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RQ

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Uses Objective I

1. Draw the path to bounce ball B off the wall andhit ball A. Mark any congruent angles made bythis path.

2. Use the miniature golf hole at the right.

a. Draw a path from the tee to the hole with abounce off wall x.

b. Draw a path from the tee to the hole with abounce off wall y.

c. Is there a direct path from the tee to thehole?


a. Draw a path from the tee to the hole using abounce off one wall.

b. Draw a path from the tee to the hole usingbounces off two walls.

c. Which path would you prefer to use? Tellwhy.

4. Use the billiard table at the right.

a. Draw a path to bounce ball Aoff wall p and hit ball B.

b. Draw another path from ball A to ball B with one bounce off a wall.

c. Draw a path to bounce ball Aoff wall m and then offwall n, and finally hit ball B.

d. Draw a path to bounce ball Aoff wall p and then off wall o,and finally hit ball B.

e. If the actual billiard table, withadditional balls in place, lookedlike the diagram at the right, whichpath from Parts a–d would be best?


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Skills Objective DIn 1 and 2, draw the indicated reflection image.

1. rm 8 r,(PQRS) 2. rq 8 rp(ABCD)

Properties Objectives F and GIn 3–5, true or false.

3. Translations do not preserve orientation.

4. If rp 8 rq(Figure N) 5 Figure M and p // q,then M is a translation image of N.

5. Under a composite of reflections over two parallel lines10 inches apart, a point is 5 inches from its image.

6. In the figure at the right, t // s, rt(ABCD) 5A'B'C'D, and rs(A'B'C'D) 5 A''B''C''D''.

a. If AA'' 5 1 cm, then CC'' 5 .

b. If AA'' 5 1 cm, then the distance between t and s is .

c. Since rt(D) 5 D, D is on line .

d. Name two segments with length equal to DC'.


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S

m

R'Q'

P'

D'C'

A'B'

AB

C

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A

C

B

A'

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Skills Objective DIn 1 and 2, draw the indicated image.

1. rp 8 rm(THE) 2. rs 8 r, (TRUE)

Properties Objectives F and GIn 3 and 4, multiple choice.

3. Rotations preserve

(a) collinearity and betweenness. (b) distance and angle measure.

(c) orientation. (d) all of the above.

4. If F is the image of G after successive reflectionsacross lines which meet at a 478 angle, then

(a) G has been rotated 478 or -478. (b) G has been rotated 948 or -948.

(c) G has been translated 47 units. (d) G has been translated 94 units.

5. In the figure at the right, m∠ COL 5 55. Considerthe rotation which maps DABC onto DA" B"C".

a. The direction of the rotation is

.

b. Give the magnitude of the rotation.

c. Name the rotation as a composite of two reflections.

d. A rotation of what magnitude e. Name the rotation in Part d as amaps DA" B"C" onto DABC? composite of reflections.

6. In each figure at the right s ⊥ p, s⊥ q. Nameeach composite of reflections as a rotation R,a translation T, both B, or neither N.

a. rq 8 rp b. rp 8 rs

c. rp 8 rp d. rq 8 rs


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Skills Objective CIn 1–3, draw and label the translation image ofthe figure determined by the indicated vector.

1. 2. 3. →CB

4. Draw and name a vector for thetranslation that maps ABCDonto A' B'C'D'.

Representations Objective KIn 5 and 6, use the vector described by the orderedpair (-7, -10).

5. Name its a. horizontal component. b. vertical component.

6. Find the image of the given pointunder translation by this vector. a. (-3, 10) b. ( p, q)

7. The image of point A under translation by the vector(-17, 33) is (-30, 29). What are the coordinates of A?

8. Draw the image of PENTA under

the translation with vector (3, -1).


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C

A

B A'

C'B'

v→

E' F'

I' G'

H'

I

F

E

G

H

w→ A'

C

F' B'A

B

E

D

F

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A'C

A

B

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B' D'

D

A'

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Skills Objective CIn 1 and 2, draw the indicated glide reflection.

1. Draw G(ABCDE) 5 T 8 rp where

T is determined by →v.

2. Draw G(PQRS) 5 rm 8 T where

T is determined by→RQ.

3. DABC is the image of DRSTunder a reflection over line ,followed by a translation.Draw a translation vector.


A

A'

EE'

D'

A

D

C

B

C' B'

pv→

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Q

Q'

S'P R

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39

4. On the grid at the right, drawH(PQRS) 5 T 8 ry-axis where T has

the vector (-3, 2).


5. Name two isometries that do not preserve orientation.

6. If you know that an isometry is thecomposite of three reflections, whatisometry might this composite be?Draw one possible situation foryour answer.

In 7 and 8, draw reflecting lines of an isometrythat is the composite of three reflections

7. and is a glide reflection. 8. and is not a glide reflection.

Uses Objective HIn 9–12, name the type of isometry that maps one letter onto the other.

9. 10. 11. 12.

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P'

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x

P

5

-5

-5

5

y

A"'A"

A'

A

A"'

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Vocabulary

1. What is another name for an isometry?

2. A treble clef is pictured at the right.

a. Draw a figure which is b. Draw a figure which isdirectly congruent to oppositely congruent tothe treble clef. the treble clef.

Uses Objective JIn 3–6, tell whether the performed task uses theidea of congruence. Explain why or why not.

3. blowing up a balloon

4. breaking a vase

5. flipping a coin

6. cutting cookies with a cookie cutter


A

Name


VocabularyIn 1–4, true or false. Inthe diagram at the right,T(ABC) 5 XYZ.

1. The side corresponding to AC is XZ.

2. The side corresponding to XY is BC.

3. The angle corresponding to ∠ A is ∠ Z.

4. The angle corresponding to ∠ XYZ is ∠ ABC.

Skills Objective A

5. Suppose r, (POST) 5 PMAN, PO 5 7 cm,OS 5 17 cm, ST 5 5 cm, and TP 5 16 cm.Which side length in PMAN is 5 cm?

6. rm 8 rn (WHY ) 5 NOT, where m and n intersect,at a 608 angle. m∠ W 5 67, m∠ H 5 30, andm∠ Y 5 83.

a. Which angle in nNOThas measure 67?

b. If NT 5 16 cm, whichside in nWHY has thesame length?

Properties Objective E

7. nBUG > nFLY. List six pairs of congruent parts.

WT

HY

N

O

n

m

A

P

O

SM

N

T

,


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Skills Objective A

1. In the figure at the right, supposeMD > DL, O is the midpoint of MD,and E is the midpoint of DL.If MD 5 28j, find each length.

a. DL b. OD c. DE d. ME

2. In the figures at the right, suppose

∠ ABL > ∠ MIS,→BE bisects

∠ ABL, and→IT bisects ∠ MIS.

If m∠ ABE 5 2.5t, find eachmeasure.

a. m∠ MIT b. m∠ MIS

Properties Objective EIn 3 and 4, name the property of congruence illustrated.

3. ∠ TOP > ∠ TOP

4. If PO > TL and TL > MN, then PO > MN.

5. You are given only that ∠ COP > ∠ TOL.Draw a possible diagram. What can youconclude about ∠ COP and ∠ TOL?

6. Do the statements PQ > RS and PQ 5 RS mean thesame thing? Why or why not?

OC L

TP

A

IL S

M

E T

B


A

C PT O

L

DO LM E

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Properties Objective EIn 1–3, multiple choice. Choose the justificationwhich allows the given conclusion.

1. If PQ 5 PT, then PQ > PT.

(a) CPCF Theorem (b) definition of midpoint

(c) definition of congruence (d) Segment Congruence Theorem

2. If →RZ is the bisector of ∠ XRT, then ∠ XRZ > ∠ ZRT.

(a) definition of angle bisector (b) Corresponding Angles Postulate

(c) CPCF Theorem (d) Vertical Angles Theorem

3. If nABC > nXYZ, then ∠ A > ∠ X.

(a) CPCF Theorem (b) Corresponding Angles Postulate

(c) definition of congruence (d) Angle Congruence Theorem

4. In the diagram at the right, points A, B, and C areon (O. OB is the perpendicular bisector of AC.Provide the justification for each conclusion.

a. AP > CP

b. OA > OC

c. OP > OP

d. ∠ APO > ∠ BPC

5. Given: r, 8 rm (ABCD) 5 EFGH, where , // m.To prove: ABCD > EFGH.

6. Given: O is the midpoint of MP. P is the midpoint of OE.

a. To prove: MO > OP.

b. To prove: OP > PE.

c. Use the conclusions of Parts a and b to prove MO > PE.

POM E

A

O

C

P B


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Skills Objective B

1. Construct an equilateral triangle 2. Construct the circle which passeswith side QL. through the points given.

Skills Objective CIn 3–5, use the figure at the right. Lines p and m are parallel.

3. Name two pairs of alternate interior angles.

____________________ ____________________

4. If m∠ 6 5 128, find as many other angle measuresas possible.

5. If m∠ 2 5 39 1 t, find the measure of each angle.

a. ∠ 3 ____________ b. ∠ 5 ____________ c. ∠ 4 ____________

In 6 and 7, use the figure at the right. Lines q and r are parallel.

6. Suppose m∠ 2 5 15x 1 4 and m∠ 3 5 11x 1 15.

x 5 ____________ m∠ 2 5 ____________

7. Suppose m∠ 5 5 20y 1 1 and m∠ 3 5 12y 1 3.

y 5 ____________ m∠ 4 5 ____________

L

Q

1

43

52q

r

82

64

53

71

p m

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Properties Objective FIn 8–10, complete the proof by givingthe argument.

8. Given: MO > TO and TE > TO.To prove: MO > TE.

9. Given: m // n and ∠ 1 > ∠ P.To prove: ∠ 2 > ∠ P.

10. Given:→SR bisects ∠ PSO and→SO bisects ∠ RSF.

To prove: ∠ PSR > ∠ FSO.

Uses Objective I

11. At the right is a diagram of a lightray traveling through a glass opticalfiber. If the ray makes a 308 anglewith the bottom of the fiber, find x, the measure of the angle at the top of the fiber. Justify your answer.

R

SPO

F

1

P

m

n

2

E

OM

T

30°

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Skills Objective C

1. m is the ⊥ bisector of AB.

a. Name 2 pairs of congruent segments.

b. If AP 5 10, then 5 10.

c. If AB 5 10, then AC 5 .

2.→RQ is the ⊥ bisector of MP. RM 5 3a,MQ 5 2b, RP 5 14, and QP 5 12.

a 5 b 5

3. Suppose ←→AC is the ⊥ bisector of XY. B is on

←→AC. AX 5 12t 1 1, AY 5 8t 1 15, and BX 5 8t.Find BY.

Properties Objective GIn 4 and 5, supply the justification for eachstep of the proof.

4. Given: rt(A) 5 C and rt(B) 5 D.

To prove: AB > CD.

0. rt(A) 5 C;rt(B) 5 D

1. rt(AB) 5 CD

2. AB > CD

A

D

C

t

B

A

X C Y

B

P

R

M

Q

A

PC m

B


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47

5. Given:←→TR is the ⊥ bisector of IA.

To prove: nTRA > nTRI.

0.←→TR is the ⊥ bisector of IA

1. r←→TR

(I ) 5 A

2. r←→TR

(R) 5 R; r←→TR

(T ) 5 T

3. r←→TR

(nTRA) 5 nTRI

4. nTRA > nTRI

In 6 and 7, complete the proof by writingan argument.

6. Given: r,(A) 5 D and r,(B) 5 E.To prove: nACB > nDCE.

7. Given: JH is the ⊥ bisector of GI.To prove: ∠ G > ∠ I.

Uses Objective I

8. Use the street map at the right. The shortestdistance from Alexa’s house (A) to the postoffice (P) is the same as the shortest distancefrom the post office to the library (L). A, P, and L are on the same street. Explain why theshortest distance from Alexa’s house to theschool (S) is the same as the shortest distancefrom the school to the library.

A D

C

,

B E

A

I

T

R

G

JH

I

A

S

P L

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1. In nABC, there is 2. Segment AC is the only one bisector unique segmentof ∠ BAC. connecting A and C.

3. There is a unique 4. It can be assumed that bisector of PE in in QUAD diagonalPENTA. QA bisects ∠ DQU.

5. In nSCA, a student wished to draw as an auxiliary

figure the ray→SL which bisects ∠ CSA and is ⊥ to

CA. Is this possible? Why or why not? If youwish, draw a picture to support your explanation.

6. In the figure at the right, how many diametersof (P are parallel to ,? Justify your answer.

Culture Objective J

7. Who was first known to suggest the uniquenessof a line parallel to a given line through a pointnot on the given line, known also as Playfair’s Parallel Postulate?

8. Non-Euclidean geometries are applied in whatwell-known physical theory?


A

A C

B

T N

EA

P

A

C

B

D

A U

D Q

A

C S

L

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P

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Skills Objective DIn 1–4, find the sum of the measures of theinterior angles of the figure.

1. scalene triangle 2. heptagon

3. decagon 4. 16-gon

5. For the figure at the right, find each measure.

a. a b. m∠ QPR

6. Find the measure of each angle of the pentagon.

a. ∠ A b. ∠ B

c. ∠ C d. ∠ E

7. The measures of the angles of a triangle are in the extendedratio 8:6:2. Find the measure of the largest angle.

8. In triangle ABC, m∠ A 5 80. The measure of ∠ B is 15 more than 9 times the measure of ∠ C. Find the measure of ∠ B and of ∠ C.

Culture Objective J

9. Why did Gauss measure the angles of the triangle whosevertices were three mountaintops?


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Skills Objective AIn 1–3, draw the symmetry line(s) for the figure.

1. 2. 3.


4. For every figure P, if r,(P) 5 P', then r,(P' ) 5 P .

5. One symmetry line for PR is the ⊥ bisector of PR.

6. Any line intersecting a circle at two differentpoints is a symmetry line for that circle.

7. If r,(ABCD) 5 PQRS, then AC > PR.

8. m and n are symmetry lines for polygonEFGHIJKL at the right.

a. Name three anglescongruent to ∠ G.

b. rn(GHLEF ) 5

c. Must LK 5 KJ? Explain your answer.

9.→PT bisects ∠ EPC.

a. r →PT

( →PC ) 5

b. What theorem justifies your answer to Part a?

Uses Objective IIn 10–12, draw the symmetry line(s) for the figure.

10. 11. 12.

BC


A

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Name


VocabularyIn 1–3, use nNPO at the right.

1. Name its base.

2. Name the base angle(s).

3. Name the vertex angle(s).

Skills Objective AIn 4 and 5, use nNPO above.

4. Draw the triangle’s symmetry line. 5. Mark a pair of conguent angles.

Skills Objective BIn 6 and 7, draw an example of the figure. Markthe congruent sides and congruent angles.

6. an isosceles triangle 7. an acute isosceles triangle

Skills Objective CIn 8 and 9, find the indicated measures.

8. m ∠ Q 5 9. m ∠ O 5 132

m ∠ P 5 m ∠ T 5

m ∠ R 5

10. nISO is isosceles with base IO.

a. If m ∠ I 5 (40 2 3x) and m∠ O 5 (5x 2 8), find x and the measure of each angle.

x 5 ∠ I 5 ∠ O 5 ∠ S 5

b. If IS 5 2z 2 2, SO 5 z 1 5, andOI 5 4z 2 6, find z and each length.

z 5 IS 5 SO 5 OI 5


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11. Draw a triangle with three symmetrylines. Mark the symmetry lines.

12. nMNO is isosceles with vertex M and MP ⊥ NO.Name all pairs of congruent angles and segments.

Properties Objective HIn 13 and 14, complete the proof by givingthe argument.

13. Given: nABL is isosceles with base AB;nBLE is isosceles with base LE .

To prove: AL > BE.

14. Given: nMIC is isosceles with vertex angle M;nHAC is isosceles with vertex H.

To prove: ∠ I > ∠ A.

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Vocabulary

1. Define each term.

a. kite

b. rhombus

c. trapezoid

d. parallelogram

Skills Objective BIn 2–4, draw a polygon satisfying thegiven conditions.

2. an isosceles 3. a pentagon that 4. a kite that is notright triangle is equiangular but a parallelogram

not regular

Skills Objective DIn 5 and 6, use parallelogram ABCD at the right.

5. If m ∠ D 5 4x, give the measure of each angle.

a. ∠ A b. ∠ B

6. If AB 5 22 and AD 5 15, find each length.

a. BC b. CD

Properties Objective GIn 7–9, use the markings on the quadrilateral togive it as specific a name as possible.

7. 8. 9.


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B

D

CA


10. Complete the argument below.

Given: Quadrilateral ABCD with AD ⊥ DC and BC ⊥ DCTo prove: ABCD is a trapezoid.

Conclusions Justifications

0. Quadrilateral ABCD;AD ⊥ DC; BC ⊥ DC

1. AD // BC

2. ABCD is a trapezoid.

11. Complete the proof by giving the argument.

Given: PO // AR and PA > OT. DOTR isisosceles with vertex O.

To prove: PORA is an isosceles trapezoid.


12. a. Draw a hierarchy relating quadrilateral,kite, square, trapezoid, and rhombus.

b. True or false. All kites are rhombuses.

c. True or false. Every square is a kite.

13. Draw a hierarchy relating quadrilateral,square, parallelogram, trapezoid, and isosceles trapezoid.

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Skills Objective AIn 1 and 2, use COAL, a kite with ends C and A.

1. Draw the symmetry line for COAL.

2. a. Draw OL . b. OL CA

c. Name two congruent angles.

Skills Objective D

3. Use kite KITE with ends I and E. If m∠ 1 5 29and m∠ KIT 5 88, find each angle measure.

4. CORK at the right is a rhombus. Find each lengthand each angle measure.


5. Use the markings to give as specific a name as possible to each quadrilateral.

a.

b.



Given: Isosceles nTRI with base TI andisosceles nTAI with base TI.

To prove: TRIA is a kite.

OR

m∠ 2

m∠ KCO

RK

m∠ 3

m∠ COR

m∠ KTR

m∠ 4

m∠ 2

m∠ KET

m∠ 3

m∠ 5

m∠ 6

m∠ 7

m∠ 4

m∠ ITE


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10 10 10

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Skills Objectives A and D

1. Suppose TRAP is a trapezoid with bases TR andAP, m ∠ P 5 45, and m ∠ R 5 93.Find m ∠ A and m ∠ T.

m ∠ A m ∠ T

2. a. Draw the symmetry lines for rectangle RECT.

b. If m ∠ REC 5 28 2 2a, find a.

3. In ABCD, m ∠ D 5 3y 1 2, m ∠ C 5 142 2 4y,m ∠ A 5 4z 1 10, m ∠ B 5 145 2 z. Find m ∠ Band m ∠ D.

m ∠ B m ∠ D

Properties Objective GIn 4 and 5, true or false.

4. Every rectangle is an isosceles trapezoid.

5. A square has four symmetry lines.



Given: MP 5 NO and m∠ 1 5 m∠ N.To prove: MNOP is an isosceles trapezoid.


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Skills Objective AIn 1–3, if the figure has n-fold rotation symmetry,find n and mark the center of symmetry.

1. 2. 3.

rectangleequilateral triangle

4. If a figure has 10-fold rotation symmetry, then theleast positive magnitude of a rotation that will mapthe figure onto itself is ? .

5. Draw a quadrilateral with the given symmetry(ies).

a. reflection and rotation b. neither reflection nor rotation

Uses Objective EIn 6–8, true or false.

6. All kites possess reflection symmetry.

7. A rectangle has reflection symmetry but not rotation symmetry.

8. All trapezoids are reflection-symmetric.

9. Name the quadrilaterals with both reflection and rotation symmetry.

Uses Objective IIn 10 and 11, the object has n-fold rotationsymmetry. Find n.

10. 11.


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ancient Celtic design Ashanti box lid, Ghana

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Skills Objectives A and BIn 1–3 draw the figure and any symmetry lines.

1. a regular triangle 2. a pentagon that is 3. a quadrilateralequilateral but not equiangular but notregular regular

In 4 and 5, locate the center of symmetry.

4. 5.

Skills Objectives C and D

6. Find the measure of each angle of a triangle formedby three adjacent vertices of a regular 15-gon.

7. Find the measure of one interior angle of a regular

a. convex octagon. b. convex 26-gon.


8. a. Name the center of rotationof the nonagon at the right.

b. m ∠ ROQ 5

c. m ∠ RQO 5 d. m ∠ ROU 5



Given: ABCDEF is a regular hexagon.To prove: nBDF is equilateral.


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Vocabulary

1. Multiple Choice. A chord of a circle is

(a) a line through the center of the circle.

(b) a segment which intersects the circle at exactly one point.

(c) a ray through two points on the circle.

(d) a segment that connects two points on a circle.

2. True or false. In a circle, the shorter the chord,the shorter the arc.

Uses Objective JIn 3 and 4, consider a round-robin match for11 teams in which games are played each week.

3. a. Mark the 11 teams on the circle at the right.

b. Draw the chord between two non-adjacentteams. Then draw all the chords parallel tothis chord. Write the first set of pairings.

c. Rotate the chords of a revolution.

Write the second set of pairings.

4. Continue until all the teams have played each other. Thenanswer the following questions.

a. How many weeks are needed?

b. How many individual games are needed?

c. How many byes are needed?

d. How many weeks would be needed for twelve teams?

5. a. Write a schedule for a round-robin tournament for 6 teams.

b. How many weeks are needed?

c. How many individual games are needed?

d. How many byes are needed?

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Skills Objective AIn 1–6, use an automatic drawer or other drawingtools. Draw the triangle with the given conditions.Then tell whether you think all triangles with theseconditions are congruent.

1. nGHI with m∠ G 5 90, 2. nCAT with CA 5 2 cm,m∠ I 5 30, and m∠ H 5 60 AT 5 2 cm, and m∠ A 5 40

____________ ____________

3. nTUP with m∠ T 5 50, 4. nLCH with LC 5 4 cm,m∠ U 5 100, and UP 5 1.50 CH 5 3.5 cm, and m∠ C 5 50

____________ ____________

5. nSMH with SM 5 4 cm, 6. nBAT with m∠ T 5 40,SH 5 4 cm, and MH 5 7 cm BA 5 3 cm, and AT 5 4.5 cm

____________ ____________

7. Mrs. Liu challenged her class to construct a uniquetriangle with sides measuring 2 cm, 3 cm, and 6 cm.How would you respond to her challenge?

A T

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H

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T

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T

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IG


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Skills Objective AIn 1–4, tell whether the conditions are sufficient toguarantee congruent triangles. Justify your answer.

1. ST 5 6 in., TU 5 4 in., m∠ T 5 110

2. m∠ S 5 25, m∠ T 5 45

3. m∠ U 5 70, m∠ T 5 46, ST 5 6.4 cm

4. ST 5 20, SU 5 40, m∠ U 5 40

Properties Objective CIn 5–8, tell whether the triangles are congruent.If yes, write a congruence statement to indicatecorresponding vertices.

5. 6.

7. 8.

9. Two triangles are shown at theright. What additional informationis needed to show congruence?

Uses Objective I

10. Triangular supports for a coldframe were constructed asshown. Why are all the supportsthe same size and shape?

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Properties Objectives D and EIn 1 and 2, fill in the blanks to complete the proof.

1. Given: m∠ PLT 5 90, m∠ LTO 5 90,and LP > TO.

To prove: nLTO > nTLP.

Argument:


0. m∠ PLT 5 90, m∠ LTO 5 90,LP > TO

1. ∠ PLT > ∠ LTO

2. LT > LT

3. nLTO > nTLP

2. Given: LU > UE and JU > UI.To prove: JE > LI.

Argument:


0. LU > UE; JU > UI

1. ∠ JUE > ∠ IUL

2. nJUE > nIUL

3. JE > LI

In 3–6, write a proof argument.

3. Given: W is the midpoint of OT and ∠ O > ∠ T.To prove: ∠ B > ∠ I.

Argument:


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4. Given: ∠ E > ∠ R and ∠ EZB > ∠ RBZ.To prove: ZE > BR.

Argument:

5. Given: (O and AB > CD.To prove: ∠ COD > ∠ AOB.

Argument:

6. Given: Isosceles trapezoid TRAP with bases PTand AR; M is the midpoint of AR.

To prove: PM > TM.

Argument:

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A

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VocabularyIn 1 and 2, redraw the figures separating thepairs of overlapping triangles. Label yourdrawings and mark the shared parts.

1. 2.

Properties Objectives D and EIn 3–5, provide the argument for the proof.

3. Given: ∠ P > ∠ T and PS > TQ.To prove: nPSR > nTQR.

4. Given: JULIO is a regular pentagon.To prove: JL > IU.

5. Given: PS > QR and ∠ PSR > ∠ QRS.To prove: ∠ 1 > ∠ 2.

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P

T

L

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RS1 2

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Skills Objective AIn 1–4, determine whether all triangles withthe same measures as the given triangle arecongruent. Justify your answer.

1.

2. a right triangle with hypotenuse12 m and one leg 5 m

3.

4.

Properties Objective CIn 5–8, if the given triangles are congruent, justifywith a triangle congruence theorem. Otherwise, write not enough to tell.

5. 6.

7. 8.


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"7

5

2110°

"

"

5"2"

15°

A CB

D

A CB

DAB > BD

A CB

DBD > AB

A CB

D

Properties Objectives D and EIn 9 and 10, write an argument to completethe proof.

9. Given: HR ⊥ HA, HR ⊥ DR, and HD > AR.

To prove: nHRA > nRHD.

10. Given: S is the midpoint of AY, ES > RS,AY ⊥ EA, and AY ⊥ RY.

To prove: EA > RY.

Uses Objective I

11. The volleyball net pictured at theright is supported by four ropes ofequal length, extending from thetops of the poles to the ground.The poles are the same length andare perpendicular to the ground.Explain why each rope makes the same angle x with the ground.

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RD

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C DG

H

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F x

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Uses Objective J

1. Use the rectangle with length twice itswidth to create a brick-patio tessellationdifferent from the one shown at the right.

2. A countertop is to be tiled with equilateraltriangles. Use the triangle at the right todemonstrate such a region.

3. Use the design at the right. Outlinethe fundamental region used tomake the tessellation.

4. Explain why a regular hexagon can tessellate a regionand a regular heptagon cannot.


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Properties Objective FIn 1–3, true or false.

1. The center of rotation symmetry for a parallelogramis at the intersection of its diagonals.

2. All parallelograms have a symmetry line.

3. In any parallelogram, two consecutive anglesare supplementary.

In 4 and 5, use parallelogram PARL .

4. Locate the center of rotation symmetryof PARL.

5. Find x and y.

x 5 y 5

6. Use parallelogram ABCD at the right.

a. If MB 5 19 and AB 5 10, find as manyother lengths as you can.

b. If m∠ DAC 5 52 and m∠ BDC 5 22, findas many other angle measures as you can.

Uses Objective K

7. Todd County in South Dakota has the approximate shape of a rectangle. Its northern border is about 40 miles long.

a. What is the length of its southern border?

b. What property of parallelogramsdid you use in Part a?

8. Woodruff Park in Atlanta, Georgia isbounded by the four streets pictured at theright. Auburn is parallel to Edgewood and Peachtree is parallel to Prior. Peachtreemeets Edgewood at an angle of 648.

a. At what angles does Auburn meet Prior?

b. What property of parallelograms isillustrated by your answer to Part a?


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P 6x + 9 A

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23 y

A

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B

M

Auburn Ave.

Edgewood Ave.

Peac

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.

Prio

r St.Woodruff

Park

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Properties Objective GIn 1–4, use the diagram. Are the markings asufficient condition for the quadrilateral to bea parallelogram? If your answer is yes,provide the sufficient condition.

1. 2.

3. 4.

5. Consider this conditional: If three angles of a quadrilateralhave equal measures, then the quadrilateral is a parallelogram.

a. Draw an instance of b. Draw a counterexamplethe conditional. to the conditional.

Uses Objective K

6. Raul carefully measured the sides of his garden. Twosides measured about 309 and two others measured about199. Is the meadow a parallelogram? Why or why not?

7. An expandable gate is constructed so that for eachquadrilateral region, AB 5 CD and AD 5 BC.

a. Explain why ABCD remains a parallelogram forany expansion of the gate.

b. Explain why m∠ B always equals m∠ D.


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100°

100°

100°

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Skills Objective B

1. a. Draw an exterior angle at each vertexof nABC at the right.

b. The sum of the anglesthat you drew in Part a is .

2. 3.

Find m∠ 1. Find m∠ 2.

4. 5. Find x and m∠ ACB.

Find m∠ 3.

Find m∠ 4. x m∠ ACB


6. Use the diagram at the right.

a. What is the relationshipbetween m∠ 1 and m∠ P?

b. What theorem justifies that relationship?

c. If m∠ 1 5 5a 2 60 and m∠ N 5 2a, write aninequality to express the possible values of a.

7. Use nRAG at the right.

a. Name the largest angle.

b. Name the smallest angle.

8. Name the sides of nABC at the right in orderof length from shortest to longest.


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A

CB

156°

22°

R

P

Q

2149°

4

61°

102° 3

(2x)°

A

CB

(5x – 3)°(6x + 19)°

D

1O

P

N

9.5

GR

A5.2

11.7

59.5°

BC

A

60.5°

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Skills Objective AIn 1–5, give the perimeter of each figure.

1. a rectangle with length 5 cm and width 3 cm

2. an equilateral triangle with one side of length 9 in.

3. a square-shaped region of land mile wide

4. a regular heptagon with one side of length 10 mm

5. a regular pentagon with side (x 1 3)

6. Pictured at the right is kite DEFG.If its perimeter is 48, what are thelengths of its sides?

Uses Objective H

7. A billboard is 1.5 times as wide as it is high. If itsperimeter is 40 meters, how wide is the billboard?

8. A tennis court is 36 feet wide and78 feet long. A fence is to be builtaround the court 10 feet from theedges of the court. How muchfencing will be needed?

9. The fences surrounding an isosceles-triangle-shapedranch have a total length of 18 km. If the triangle’sbase is 4 km, find the lengths of the other sides.

10. Flowers are to be placed at 1-foot intervals arounda rectangular garden. If the garden is 4 yards wideand 16 yards long, how many flowers will be needed?

12


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Skills Objective C

1. Rectangle RECT has dimensions 9 feet and 15 feet.

a. Find Area(RECT ) in square feet.

b. What are the dimensions of RECT in yards?

c. Find Area(RECT) in square yards.

2. Find the area of a square with side length inch.

3. A square has a perimeter of 20x.

a. Give an expression for the area of the square.

b. Evaluate the area for x 5 3.

4. The area of a rectangle is 455 square centimeters andits length is 26 centimeters. Find its width.


5. Draw three different rectangles with perimeter of30 units. Find the area of each.

Uses Objective I

6. How much carpet is needed to cover the floorof a room that is 12 feet by 16 feet?

7. How many 8-yd-by-25-yd tarpaulins will be needed tocover a soccer field which measures 50 yd by 120 yd?


8. Find the perimeter and area of JKLMNO.

perimeter

area

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A

(-2, 4)

L (2, 0)

M(2, -2)

N(5, -2)

K(-2, 0)

x

yJ

(5, 4)O

7.5 un.

7.5 un.

56.25 un.25 un.

10 un.

50 un.2 3 un.12 un.36 un.2

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Skills Objective BIn 1–3, use the method in this lesson to estimatethe area of the irregular region. The side lengthof one small square is given.

1. 1 mile 2. .2 km 3. 50 yd

4. Estimate the area of the kite.

a. Use the left-hand grid.

b. Use the right-hand grid.

side of squares 5 2 cm side of squares 5 1 cm

c. Which answer is more accurate? Explain.

5. An irregularly shaped region appears on a computerscreen. 150,000 pixels are contained within the regionand 7,200 pixels are on the region’s boundary.

a. What is the area of the region (in pixels)?

b. If the entire screen is composed of 640 rows and480 columns of pixels, what percent of the screenis covered by the region?

c. Find the area of the region with 150,000 pixels onthe region’s boundary and 7,200 pixels containedwithin the region.

d. What percent of a screen composed of 640 rowsand 480 columns of pixels is covered by this region?


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Skills Objective CIn 1–3, find the area of nABC.

1. 2. 3.

AD 5 9 cm, DB 5 5 cm,DC 5 2.5 cm Area (nACD) 5 60 m2

4. If Area (nDEF) 5 1134, find GE.


5. Explain how the formula for the area of a right triangleis related to the formula of a rectangle.

Uses Objective I

6. On the boat shown at the right, the height of the mastis 18 feet and the length of the boom is 14 feet.

a. How much material is needed to construct a triangular sail for the boat?

b. If the same amount of material is used for a sail for a boat with a 21-footmast, whatis the maximum length for its boom?


7. A triangle has vertices (-3, 3), (2, 7),and (5, 3).

a. Draw the triangle on the grid atthe right.

b. Find the area of the triangle.


A

B

A

DCx 5x

12B

CAD

3 BC

A

4 5

E

D

19GF

44

14’

18 ’

y

-2 x-2

-4-6

2

4

6

8

2 4 6

Name


Skills Objective CIn 1–5, use the information in the drawingto find the area of the largest trapezoid.

1. 2. 3.

4. 5.

6. The area of a rhombus is 232 cm2, and its perimeteris 64 cm. Find the length of its altitude.

7. A trapezoid has an area of 120 in2. Its altitudemeasures 8 in. Give a possible pair of lengths forthe bases of the trapezoid.


8. Of parallelograms ABCD, EFCD, andGHCD, which has the greatest area? Howdo you know?


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6 4.9

5.7

5.8 2.1

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B A F E H G

C D

Uses Objective I

9. The Canadian province ofSaskatchewan is shaped roughlylike an isosceles trapezoid whosedimensions, according toSaskatchewan’s Office of CentralSurvey and Mapping, are given onthe map at the right. Estimate thearea of Saskatchewan.

10. A lawn-maintenance person canmow about 9,000 square feet ofgrass per hour. At this rate, howlong would it take for this person tomow the plot of land at the right?

Representations Objective KIn 11 and 12, find the area of the region.

11. 12.

76

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773 km

277 km

126 ft149 ft210 ft

290 ft

y

x

-10

10

-10

20

30

40

10 20 30 40 50

y

x-1 2 4 6

1

3

5

7(3, 6)

(7, -1)(1, -1)

(-1, 6)

Name


Skills Objective DIn 1–3, find the length of the missing side andthe area of the triangle.

1. 2. 3.

Skills Objective EIn 4 and 5, could the numbers be the lengthsof sides of a right triangle?

4. 60, 80, 100 5.

6. Find the perimeter and the area of a 7. Find the perimeter and the area of right triangle with a hypotenuse of a rhombus with diagonals10 mm and one side 6 mm long. measuring 16 in. and 30 in.

perimeter perimeter

area area

Uses Objective H

8. The infield of a baseball diamond is a square-shaped region 90 feet on a side. When a center-fielder throws a ball to home plate from 100 feetbehind second base, how long is the throw?

9. Ramona found a letter on the corner of a rectangular park and decided to put it in amailbox which was on the opposite corner of the park. The perimeter of the park is 344 m, and it is 10 m longer than it is wide. How much farther is it for her to walk along the sidewalk than to walk directly to the mailbox?

Culture Objective L10. Identify the cultures that knew of the Pythagorean

Theorem more than 2000 years ago.

Ï3, Ï4, Ï5


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Skills Objective F

1. Multiple choice. Which expression shows the exactcircumference of a circle with a diameter of 15 km?

(a) 7.5πkm (b) 47.1 km (c) 30πkm (d) 15πkm

2. Give the circumference of a circle with a diameterof 12 feet

a. exactly. b. to the nearest foot.

In 3–5, find the radius of a circle whose circumference is given.

3. 118π 4. 8.75 ft 5. 14x mm

In 6–8, find the length of an arc with the givenmeasure on a circle whose radius is 18.

6. 180˚ 7. 120˚ 8. 72˚

9. In the circle at the right, OB 5 3.5 andCHB 5 2.75. Find mCHB.

Uses Objective J

10. The earth’s orbit can be approximated by a circlewith a radius of about 93,000,000 miles. What isthe circumference of this circle?

11. The gold for a certain wedding band costs $12per millimeter. How much more expensive is thewedding band for someone whose finger has adiameter of 17 mm than for someone whose fingerhas a diameter of 14 mm?


A

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O

H2.75

3.5

Name


Skills Objective FIn 1–3, estimate to the nearest tenth the area of the circle described.

1. a radius of 6 inches 2. a circumference of 3. a diameter of cm18πunits


4. Which has the greater area, a circle with a diameter of 4 feetor a square with a side of 4 feet? Justify your answer.

Uses Objective J

5. Suppose a baseball field is shaped like aquarter-circle with a radius of 350 feet asshown at the right. What is the area of theoutfield (the shaded region)?

6. A cellular phone can transmit a call anywhere within a 50-mile radius from the transmission point.What is the area of the transmission region?

7. A CD with a diameter of 12 cm is packed in a square casewhich measures 13 cm along a side. How much space isaround the CD?

2Ï5


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Vocabulary

1. State the converse of the Flat Plane Assumption.

2. Define intersecting planes.

Properties Objective FIn 3–5, state whether the figure is contained in aunique plane. State the Point-Line-Plane PostulateAssumption which justifies your answer.

3. all of Euclidean space

4. three collinear points

5. an isosceles triangle

In 6–9, match the situation with the Point-Line-PlanePostulate Assumption it most closely illustrates.

(a) Unique Line Assumption (b) Number Line Assumption

(c) Dimension Assumption (d) Flat Plane Assumption

(e) Unique Plane Assumption (f) Intersecting Planes Assumption

6. the stability of a tripod

7. on a piece of notebook paper, there are points other than those on the top edge

8. two walls that meet at a corner in a room

9. on a flat green, a putt along only one path willget a golf ball from its lie to the hole


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Skills Objective AIn 1–4, draw the figure.

1. a line which intersects a plane but 2. a segment perpendicular to both ais not perpendicular to the plane plane and a line parallel to the plane

3. a line which is perpendicular 4. a dihedral angle with measure 65to two planes


5. Use the figure at the right, in which , isperpendicular to plane R at O, and m andn are in plane R.

a. is , ⊥ m? Explain your answer.

b. is m ⊥ n? Explain your answer.

6. Provide an argument for the proof.

Given: Points A, C, D and E lie in plane X.AC is a bisector of DE. PB is ⊥ toplane X.

To prove: DP 5 EP.


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Skills Objective AIn 1–4, sketch the indicated surface.

1. a right cylinder 2. a right rectangular prism

3. a right triangular prism 4. an oblique pentagonal prism

Skills Objective D

5. Consider the triangular prism drawn at the right inwhich CA ⊥ AB.

a. Find FE.

b. What is the area of each lateral face?

6. Use the oblique cylinder at the right.

a. Find the area of its base.

b. Find its height.

Uses Objective HIn 7–9, tell which 3-dimensional figure most resemblesthe real-world object. Be as specific as you can.

7. a can of soup 8. a tent 9. a row of CD boxes thatleans in one direction


A

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EB

A D

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3

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1. a right cone whose height is 2. a right hexagonal pyramidshorter than the radius of its base.

3. an oblique cone whose smaller 4. a truncated square pyramidangle to the base has measure 60

Skills Objective D

5. The right square pyramid at the right has lengthsas marked.

a. Find its height.

b. Find the area of one lateral face.

6. In the regular pentagonal pyramid at the right,AC 5 9 and DC 5 8.

a. Find the perimeter of its base.

b. Find its slant height.


7. a flower pot

8. the Transamerica Building

9. a stalactite


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Skills Objective A

1. Draw a hemisphere with a horizontalbase and a vertical plane section of it.

Skills Objective BIn 2–5, sketch the plane section and name its shape.

2. parallel to the base 3. parallel to the bases ofof a square pyramid an oblique cylinder

4. perpendicular to 5. neither parallel tothe base of a right nor intersecting thecone and through bases of a regularthe vertex hexagonal prism

Skills Objective D

6. The radius of a sphere is 14.4 in. What isthe area of a great circle of the sphere?

7. A plane section is formed when a plane cutsperpendicularly through the diameter of thebase of the right cylinder pictured at the right.What is the area of the plane section?

Uses Objective HIn 8 and 9, identify both the 3-dimensional figureand the kind of plane section described.

8. the middle layer of icing in a round birthday cake

9. a florist’s oblique cut through a flower stem


A

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Name


Properties Objective GIn 1–3, a figure is given. a. Tell if the figure has bilateral symmetry. b. If so, give the number of symmetry planes.

1. 2. 3.

right square pyramid coffee mug egg

a. a. a.

b. b. b.

4. A regular prism which has a hexagon for a base hashow many symmetry planes?

5. Draw a prism with no planes of symmetry.

6. Draw a pyramid with one plane of symmetry.


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Skills Objective CIn 1–3, a figure is given. Sketch views of the figurefrom a. the top. b. the front. c. the right side.

1. 2. 3.

a. a. a.

b. b. b.

c. c. c.

Skills Objective EIn 6 and 7, name a surface with these views.

6. 7.

8. Use the given views ofthe building.

a. How tall in stories isthe building?

b. How long in sections is the c. Sketch the shape ofbuilding from front to back? the building.

DO

GS

LIFE ON

EARTH

AN

IMA

LS

CATS


A

top side fronttop side front

topsidefront

L R F B

L R

Name


VocabularyIn 1–3, give the number of faces, vertices andedges of the polyhedron. Describe the faces.

1. regular tetrahedron

2. regular dodecahedron

3. regular icosahedron

Representations Objective JIn 4–6, sketch the figure that can be made fromthe given net.

4. 5. 6.

7. Draw a net for an oblique rectangular prism.

8. Draw a net for a standard die pictured at the right, makingsure that the numbers of dots on opposite faces add to 7.


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d dπd2

πd2

πd2

πd2

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Uses Objective I

1. Draw a map with six regions that 2. Draw a map with six regions thatcan be colored with two colors. needs four colors to be colored.


3. Explain one problem of using a Mercator-projection map.

4. Which of the A-B-C-D properties does the map below preserve?


A

Name


Skills Objective AIn 1 and 2, give the surface area of the figure.

1. a right triangular prism with a 2. a right cylinder with circumferencebase of side lengths 3 cm, 4 cm, 2πinches and height 1 inches.and 5 cm, and a height of 4 cm

3. Refer to the right cylinder at the right.

a. Find its lateral area.

b. Find its surface area.

4. Refer to the box at the right.



Uses Objective H

5. a. A paint roller is 12 inches long andhas a radius of inch. What is itslateral area?

b. Suppose the paint roller has a radiusof inch and is 18 inches long.What is its lateral area?

6. A gift box measures 40 in. by 28 in. by 12 in. Can thebox be completely covered by a 30-ft2 roll of wrappingpaper? Why or why not?

12

34

12


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10cm

4cm

8x

6x

5x

34

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Skills Objective BIn 1 and 2, give the surface area of the figure.

1. a right cone with a slant height 2. a right square pyramid with a slantof 12 and a radius of 6 height of 5 and a height of 4

3. Find the lateral and surface areas of the regularpyramid shown at the right.

4. The slant height of a regular square pyramidis 17 m and its lateral area is 544 m2.What is the side length of its base?

5. Find the lateral area of a right cone whoseheight is 15 m and whose radius is 8 m.

Uses Objective H

6. How much paper would be needed to constructa birthday hat which is in the shape of a rightcone with radius of 2.5 inches and a slant heightof 5 inches?

7. How much wood is needed to replace the entireroof of the gazebo at the right which is in theshape of a right pyramid on a regular octagon?


A

5050

60

8 ft

6 ft

Name


Skills Objective AIn 1–3, give the volume of the box with thegiven dimensions.

1. 4, 4, 4 2. 8 cm, 11 cm, 18 cm 3. 5.1 in., 8.8 in., 1 ft

4. Find the volume of a cube with side length p.

5. Find the volume of a cube with side length ( p 1 1).

In 6 and 7, give the volume and the surface areaof the box.

6. 7.

V 5 V 5

S.A. 5 S.A. 5

Skills Objective CIn 8–10, give the cube root of the given numberto the nearest tenth.

8. 343 9. 3.375 10. 2,000

11. The volume of a cube is 60 cubic centimeters. Whatis the length of an edge, to the nearest tenth?

Uses Objective I

12. An air-conditioner manufacturer claims that a certainmodel can cool any room with volume no greater than2,000 cubic feet. Can it cool a room that measures14 feet by 12 feet by 9 feet? Explain your answer.

13. The volume of a box is 12,960 cm3. It is 24 cm longand 18 cm wide.

a. What is its height? b. What is its volume in cubic millimeters?


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1. The dimensions of a box are x, 2x and 4x. Whenall dimensions are multiplied by 3,

a. what happens to its volume?

b. what happens to its surface area?

2. A box has dimensions 5 in., 8 in., and 10 in.

a. What are the volume and surface area of this box?

V 5 S.A. 5

b. What are the volume and surface area if each dimension is multiplied by 4?

V 5 S.A. 5

3. The length and width of the bottom of a box are eachmultiplied by . How will its lateral area change?

Representations Objective JIn 4–6, expand.

4. r (2r 1 1)(3r 1 4) 5. (2y 1 2)(3y 1 9) 6. (x 1 2)3

7. Find the area of the rectangle. 8. Find the volume of the box.

14


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aca

35

8

x

zy

Name


Skills Objective AIn 1–3, calculate the volume of the figure.

1. 2. 3.

4. What is the radius of an oblique cylinder that has a volumeof 722πmm3 and a height of 8 mm?

5. A right cylinder and a right prism with a square base eachhave a volume of 2205 cm3 and a height of 20 cm. Whichis greater, the diameter of the base of the cylinder or a sidelength of the base of the prism? Justify your answer.


6. Does Cavalieri’s Principle apply to the prismsat the right? Explain why or why not.

Uses Objective I

7. Which holds more water, a 7-inch tall cylindricalglass with a diameter of 2.5 inches or a 4-inch tallcylindrical glass with a diameter of 3.5 inches?

8. A horse trough is shown at the right.If it is filled to the top with grain,how many cubic feet of grain willthe trough hold?

Questions on SPUR Objectives

See pages 617–619 for objectives.10-5A

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8 8

4

17''

6'

12''

7''

13'' 13''

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1. When deriving formulas for cones and cylinders fromthe basic formulas for conic and cylindric surfaces, what circle formulas can be used?

a. B 5 b. p 5

2. For each figure below, write a specific surface-area formula.

right cone right cylinder

a. b.

regularrectangular prism rectangular prism

c. d.

3. For some figures, the formula for lateral area is L.A. 5 ,p.

a. Draw two different figures b. Draw a figure for whichfor which this is true. this is not true.

4. Give the formula for the surface area of a squarepyramid whose slant height is 3x and whose basehas a side length x 1 2.

5. Multiple choice. The general formula for the lateral area ofa right conic surface is L.A. 5 ,p. Which expression givesthe specific lateral area of a cone?

(a) L.A. 5 πrh (b) L.A. 5 (2πr 2h)

(c) L.A. 5 πr, (d) L.A. 5 (2πr , )12

12

12

12

12

12



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h,

s

Name


Skills Objective BIn 1–5, find the volume of the figure.

1. square pyramid 2. right cone 3. rectangular pyramid

4. oblique pyramid with 5. an oblique cone whose base has atriangular base circumference of 4π

6. A regular square pyramid has a base with sides of length 7.5 m. If its volume is 225 m3, what is its altitude?

Uses Objective I

7. The sharpened end of a pencil has the shape of aright cone. The diameter of the base is 7 mmand the height is 22 mm. What is the volume ofthe whole pencil point?

8. A structure is composed of a cube-shaped baseof edge length 16 ft and a pyramid-shaped roof.In order for the roof to have the same volume asthe base, what must be the altitude of the roof?



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6 11

32 20

18

23

24

10

13.5

722

16 ft

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Skills Objective DIn 1–3, draw the sphere with the given dimensionand find its volume.

1. radius 5 1 cm 2. diameter 5 2.5 cm 3. diameter 5 0.75 in.

In 4–6, give the radius of the sphere with the given volume.

4. 36π 5. 420π 6. 1234 ft3


7. The diameter of Saturn is about 10 times that of Venus.How do their volumes compare?

Uses Objective I

8. A tank full of 15,000 in3 of air is to be used toinflate a beach ball with diameter 20 in., abasketball with diameter 9.5 in., and a volleyballwith diameter 8 in. Does the tank containenough air to fill all of the balls?

9. A globe with a 16-cm radius just fits into a cube-shaped box. The rest of the box is filled withpacking material. What is the volume of thepacking material?



Name


Skills Objective DIn 1–5, find the surface area for the sphere withthe given dimension.

1. radius 5 4 in. 2. diameter 5 0.28 km

3. circumference of a great circle 5 22πcm

4. circumference of a great circle 5 16.44 m

5. volume 5 288πin3

6. A sphere has a surface area of 722 ft2. Find its radiusto the nearest hundredth of a foot.


7. The diameter of Uranus is about 4 times that of Earth.How do the surface areas of the two planets compare?

8. If a sphere’s radius shrinks to of its original size,what happens to its surface area?

Uses Objective H

9. A Hall-of-Fame baseball player wants to bronze thelast home-run ball he hit. The radius of a ball is1.5 in. Find the area of the surface to be bronzed.

10. A concert shell, in the shape of a quarter of a spherewith a radius of 80 feet, is to be painted. If one gallonof paint covers 400 square feet, how much paint isneeded to cover the outside of the shell?

11. The planet Mercury has a radius of about 2400 km.If a surveying satellite can take a picture whichcovers an area of 30,000 km2, what is the leastnumber of pictures that will need to be takento completely survey the surface?

14



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Properties Objective DIn 1–3, use both given statements. a. What(if anything) can you conclude? b. What law(s)of reasoning have you used?

1. (1) If a triangle has two congruent sides, it is anisosceles triangle.

(2) In nABC, AB > BC.

a.

b.

2. (1) If a 5 4, then b 5 7.(2) If b 5 7, then c 5 9.

a.

b.

3. (1) If quadrilateral PQRS is a square, it is alsoa rectangle.

(2) Quadrilateral PQRS is a rectangle.

a.

b.

Uses Objective HIn 4–6, use all the given statements. What(if anything) can you conclude?

4. (1) Maria practices the cello on every day whose name begins with the letter T.

(2) Today is Thursday.

5. (1) Sam eats tiramisu every year on his birthday.(2) Last Tuesday, Sam ate tiramisu.

6. (1) If the Central High Cougars win Friday’s game,they will win the championship.

(2) If Jaime pitches for Friday’s game, the Cougarswill certainly win the game.

(3) Jaime will pitch on Friday night.


A

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Properties Objective DIn 1 and 2, use both given statements. a. What (if anything)can you conclude? b. What law(s) of reasoning have you used?1. (1) If m∠ B is 90, then nABC is a right triangle.

(2) nABC is not a right triangle.

a. b.

2. (1) If 4 1 a 2 5 9, then a 5 6 . (2) a < 0

a. b.

Properties Objective EIn 3 and 4, a statement is given. a. Write itsconverse. b. Write its inverse. c. Write itscontrapositive. d. If the original statement istrue, which of b, c, and d are also true?3. In nLMN, if MN 5 5, then LM 1 LN > 5.

a.

b.

c.

d.

4. If it is cloudy, then it is raining.

a.

b.

c.

d.

Uses Objective HIn 5 and 6, use both given statements. What (if anything)can you conclude using the laws of reasoning?5. (1) Antonio lives in Stockholm.

(2) Stockholm is not located in Italy.

6. (1) A dog has four paws.(2) My pet has four paws.

Ï5


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Properties Objective D

1. If statement A or statement B is true, and statement Bis not true, then ? must be true.

2. If nRST is neither isosceles nor equilateral, what can you conclude?

3. Use all the given statements. a. What (if anything) can youconclude? b. What law(s) of reasoning have you used?

(1) WXYZ is either a rectangle or a trapezoid.(2) A quadrilateral is a rectangle if and only if it has four right angles.(3) WX is not perpendicular to XY.

a.

b.

Uses Objective H


Kevin, Mel, and Ken each ate a different breakfast, consisting of eggs, cereal, or a bagel.

(1) Mel never eats eggs.(2) Kevin ate either cereal or a bagel.

a.

b.

5. Chris, Becky, Rob, and Jim are putting on a play andthey each have a different role from among these four: theScarecrow, the Lion, the Tin Man, and Dorothy. From theclues below, determine who plays which part.

(1) The scarecrow is not Becky, Rob, or Chris.(2) Chris is very good friends with Dorothy and the Lion.(3) Rob gave a costume to Dorothy.


A

Name


Properties Objective DIn 1 and 2, statements p and q aregiven. Are p and q contradictory?Explain your answer.

1. p: EF 5 FG q: EG 5 2EF

2. p: circumference of (D 5 2πx q: area of (D 5 4πx2


3. Write an indirect proof to show that B is not thereflection image of A over line ,.

4. Write an indirect proof to show that Þ .

Uses Objective H

5. A softball team needs to pick a captain and an assistant captain. The captain must be either a junior or a senior. Tonya, Jodi, Jennifer, and Sarah all would like to be either captain or assistant captain. Jodi and Jennifer are the only sophomores. There are no freshmen on the team. Sarah will not serve unless Jodi also serves and vice versa. Use an indirect proof to show that Tonya cannot be the assistant captain.

9756Ï3


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1. If G 5 (6, 3), E 5 (4, -1), R 5 (2, 7), andM 5 (-2, 3), use an indirect proof to

show that ←→GE is not parallel to

←→MR.


2. Prove that the quadrilateral with vertices P 5 (19, 13),Q 5 (15, 10), R 5 (18, 6), and S 5 (22, 9) is a rectangle.

3. Prove that nEMG with vertices E 5 (-1, 9), M 5 (6, 7),and G 5 (4, 0) is a right triangle.

Representations Objective KIn 4–6, draw the figure in a convenient locationon the coordinate system.

4. an isosceles right triangle 5. a parallelogram 6. a square


A

(0, 0)

(0, a)

(a, 0)

y

x

(-c, b) (a, b)

(-a, -b) (c, -b)

y

x

(a, a)

(0, 0)

(0, a)

(a, 0)

y

x

2

-2-2 2 6

6

y

xE = (4, -1)

G = (6, 3)

R = (2, 7)

M = (-2, 3)

Name


Skills Objective AIn 1 and 2, find the distance between the given points.

1. (4, -3) and (-2, 6)

2. (1.5, 3.9) and (-0.4, -9.2)

3. What is the radius of the circleat the right with center D?


4. Use the figure at the right. Write anindirect proof to show that ABCD isnot a kite.


5. nXYZ has vertices X 5 (85, 150), Y 5 (115, 110),and Z 5 (125, 180). Prove that nXYZ is isosceles.

Uses Objective I

6. Gary Ozawa gave you the following directions to get fromschool to his home: Go six blocks east, then 15 blocks north,then 4 more blocks east, then 1 block south, and finally1 block west. By air to his home, about how many kilometersis it, if 5 blocks is about 1 kilometer?


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y

x

C (9, 11)

D (5, 6)

y

x

B (6, 10)

C (0, 2)

D (-3, 0)

A (-2, 4)

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Representations Objective JIn 1–5, write an equation for the circle satisfyingthe given conditions.

1. radius 6, center (-5, 2) 2. radius 14, center at the origin

3. diameter 20, center (45, 57) 4. radius 2.5, center (h, k)

5. center (4, -10), containing point (12, -10)

6. a. On the grid at the right,draw the circle with radius 7and center at (0, 4).

b. Name 2 points on the circle.

c. Write an equation forthis circle.

In 7–10, an equation for a circle is given.a. Give its center. b. Give its radius.c. Name two points on the circle.

7. (x 2 8)2 1 ( y 1 3)2 5 169 8. x 2 1 (y 2 5)2 5 272.25

a. a.

b. b.

c. c.

9. (x 2 15)2 1 ( y 2 9)2 5 60 10. 86 5 x 2 1 y 2

a. a.

b. b.

c. c.


A

(0, 4)

y

x5-5

5

-5

Name


Skills Objective AIn 1 and 2, determine the coordinatesof the midpoint of the given segment.

1. a segment with endpoints (7,-4) and (-1, 9)

2. a segment RS at the right

3. Find the midpoint of MZ, given thatZ 5 (5, 1) and that M is the midpoint of thesegment with endpoints (14, -3) and (28, 17).

Skills Objective B

4. In nEFG at the right, A and B are midpoints of EG andFG, respectively.

a. If EF 5 12 cm, AB 5 .

b. What other relationship exists between EF and AB?

5. G, H, and I are midpoints of the sides of nJKL asshown at the right. If GI 5 7 cm and JL 5 26 cm, give all other segment lengths that can be found.


6. Prove that in parallelogramWXYZ, the midpoints of thediagonals coincide.


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2

-2-2 2 6

4y

xS

R

8-4

A

G

B

FE

I

K

LJ

H G

y

x

X (0, 8) Y (10, 8)

W (–3, 2) Z (7, 2)

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Skills Objective C

1. a. Plot A 5 (4, -2, 1) and B 5 (-2, 5, 3)on the grid at the right.

b. Give the coordinates of the midpointof the segment joining A and B.

c. Find the distance between A and B.

Uses Objective I

2. A small closet is 160 cm tall, 80 cm deep, and 100 cm wide.Will a pair of skis which are 200 cm long fit in the closet?Explain your answer.

Representations Objective J

3. A sphere has equation (x 1 3)2 1 ( y 2 6)2 1 ( z 1 2)2 5 289.

a. What is the center of the sphere?

b. What is its radius?

c. Give the coordinates of two pointson the sphere.

4. Write an equation for the sphere graphed at the right with center (0, 0, 3).


A

x

B

A

z

y

(0, 0, -3)

(0, 0, 9)

z

y

x

Name


Representations Objective GIn 1–3, quadrilateral WXYZ is graphed at the right. W 5 (-4, -6),X 5 (-5, 4), Y 5 (2, 6), and Z 5 (5, 1).

1. List the coordinates of the vertices ofthe image of WXYZ under S1.5 andgraph it.

2. List the coordinates of the vertices of S (WXYZ) and graph it.

3. Give the coordinates of the image of Xunder Sk.

In 4–6, Let M 5 (10, 2), N 5 (7, 12), and O 5 (-1, 6).Let S4(nMNO) 5 nM’N’O’.

4. M’ 5 N’ 5 O’ 5

5. Verify that the slope of MN equals the slope of the line through M’ and N’.

6. Use the distance Formula to verify that N’O’ 5 4 NO.?

34


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W

W’

Z Z’

Y

Y’

XX’

W’’

Z’’Y’’X’’

5

-5

-5

5S1.5

S 34

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Skills Objective A

1. Draw the image of ABCD under a size change with center E and magnitude 2.

2. Draw the image of nJKL under a size change with center Kand magnitude 0.7.

Properties Objective C

3. For the figure at the right, use aruler to determine the center Cand scale factor k for the sizetransformation represented. The image is shown by the dashed line.

4. In the figure at the right, X’Y’Z’ is asize-change image of XYZ.

a. Is this size change anexpansion or a contraction?

b. If X’Y’ = 10 and XY = 8, findthe magnitude k of the size change.

c. Use the value of k in Part b to find YZ if Y’Z’ 5 6.4.

L

J

J’

K = K’L’

D

A

B’

C’C

B

A’

D’

E


A

C

X

Y

Z

X’

Y’

Z’

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Skills Objective AIn 1 and 2, draw the image of the figure underthe size change with center C and magnitude k.

1. k 5 2 2. k 5 .6


3. An architect’s original sketch of a building has length.8 m and height .5 m. The design is changed so that onthe sketch the new length is .32 m.

a. What is the scale factor of the contraction?

b. What is the new height of the building?

c. Find the area of the front face of each sketch of the building.

original new

d. The area of the original sketch of the building is how many times the area of the new sketch of the building?

4. At the right, nLMN is a size-transformationimage of nIJH with center O. If OH 5 10,JH 5 7, ON 5 25, and m ∠ LMN 5 32, find

a. the magnitude ofthe size change.

b. m∠ HJI.

c. MN. O HN

M

J

IL

C

C


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Skills Objective B

In 1 and 2, nBCD is the image of nEFGunder a size change.

1. If BC 5 2.8, EF 5 5.6, and CD 5 5.2,find FG.

2. Multiple choice. Whichequation is a proportion?

(a) (b)

(c) (d)

In 3 and 4, VIDEO is the image of ACTUL under a size change.

3. If DE 5 12, EO 5 9, and UL 5 15,find TU.

4. If VO 5 12, AL 5 20, and CT 5 11,find the length of another segment.

Uses Objective E

5. If 100 sheets of paper cost $1.19, about how muchwill 250 sheets cost?

6. If p pounds of carrots cost d dollars, at that rate,how much will q pounds of carrots cost?

7. One law of physics states that the amount of pressure that a gas exerts is proportional to the temperature of the gas in Kelvins (K), assuming a constant volume. If a gas exerts 16 lb/in2 of pressure at a temperature of 298 K, how much pressure would the gas exert at 278 K?

BCCD 5

BDFG

CDGF 5

BCFE

BDEF 5

BCEG

BCEF 5

BDFG


A

V D

O E

A T

I

UL

C

B

D

C

E

G

F

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Skills Objective B

1. TILE ; WALK, with sides and measuresas indicated. Find as many missinglengths and angle measures as possible.

In 2 and 3, nCBA , nONM. The ratio ofsimilitude is 2.35.

2. If NM 5 7.8, give a length of nCBA.

3. If m ∠ C 5 59, find the measure of another angle.

4. In the figure below, rectangle EFGH is similar torectangle LMJK, JK 5 16, and EF 5 11.

a. What is a ratio of similitude?

b. If EH 5 6, find MJ.

Uses Objective E

6. A Chinese tapestry is 16 feet high and 28 feet long.A reproduction of the original is 10 feet long. Howhigh is the reproduction?

7. A model car is similar to its original. The model is20 cm long and has wheels which are 2 cm in diameter. If the original car is 5 m long, what is the diameter of the actual wheel?

L M

K J

E F

H G


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K

L

E

TI

955°

5

11

9

112°

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1. Suppose two figures are similar and their lengths are inthe ratio 7 to 1. What is the ratio of their volumes?

2. Two similar isosceles trapezoids have areas of 40 and120 square units. If the longer base of the larger trapezoidhas a measure of 14 units, what is the measure of thecorresponding base of the smaller trapezoid?

3. Two triangular prisms are similar with ratio ofsimilitude 0.8. If the larger prism has volume250 cubic centimeters, find the volume of theother prism.

4. Two similar rectangles have perimeters of 76 ft and114 ft, respectively. What is the ratio of their areas?

Uses Objective F

5. The model of a cylindrical satellite is similar to an actualsatellite. If the model has a base area of 1.65 squaremeters and the model is the actual size of the satellite,

what is the base area of the actual satellite?

6. The floor areas of two similar gymnasiums are14,400 square feet and 6400 square feet. If the lengthof the floor in the smaller gymnasium is 128 feet, what isthe length of the floor in the larger gymnasium?

7. Two similar teddy bears’ heights are 50 cm and 20 cm.What is the ratio of their volumes?

8. Stop signs are shaped like regular octagons. A regulationstop sign has sides 32 cm long, while the stop sign for a snowmobile trail has sides 8 cm long. What is the ratioof the areas of the two stop signs?

110


A

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Uses Objective F

1. Two similar ancient Egyptian vases are 1 ft and 2 ft tall.If it takes 1 pint of paint to restore the larger vase, howmuch paint would the smaller vase require?

2. A bakery sells bread based on total volume. A loaf whichis 50 cm long sells for $1.69. What is the cost of a similarloaf of bread 30 cm long?

3. The model of a 350-foot-tall building is 2 feet tall.

a. What is the ratio of the base areas of the building andthe model?

b. If the area of the base of the model is 20 square inches,what is the area of the base of the actual building?

4. A class wants to have a giant pizza for a party. They estimatethey will need a pizza with a 32-inch diameter. Suppose thelocal pizzeria agreed to make this giant pizza. If a 14-inch pizza costs $12.99, how much should the class expect topay for a similar 32-inch pizza?

5. Marta and Maggie do yard work to make money.They tell their employers that they charge $10 forevery 300 square feet of lawn. A new neighbor has a corner lot pictured at the right. Marta thinks they willearn $125, but Maggie says they will earn only $10. Who is correct? Justify your answer.

75 ft

100 ft125 ft


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Properties Objective FIn 1–4, determine whether or not the triangles are similar. If they are similar, write a similarity statement using the correct order of vertices. Justify your answer.

1. 2.

3. nEFG: EF 5 , FG 5 , GE 5

nBCD: BC 5 12, CD 5 35, DB 5 42

4.nTOP: TO 5 28, OP 5 45, PT 5 56nBAT: BA 5 81, AT 5 100.8, TB 5 50.4

In 5 and 6, the triangles are similar with corresponding sides parallel. a. Find the ratio of similitude. b. Write a similarity statement using the correct order of vertices.

5. 6.

a. a.

b. b.

43

76

25

4

343

BU

L

4.5 2

6R

A

P6.5

5.5

6

J

I

H9.1

8.4

7.7

K

M

L


A

25

410

15

6

A

M

B N C

4.8

2.9

3.5

5.8 PO

R SQ

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Properties Objective FIn 1–4, determine whether the triangles are similar.If so, what triangle similarity theorem guaranteestheir similarity?

1. 2.

3. 4.

5. In the figure at the right, ←→XY //

←→ST,

Prove that nSTC , nYXC.

Uses Objective H

6. A tree casts a shadow that is 8 m long atthe same time a meter stick casts a shadow40 cm long. How tall is the tree?

7. A freeway ramp is 10 ft high after 200 ft.How high is the highest point of the ramp ifthe ramp is 1500 ft in length?

24

13.2

2

1.252

7

14

612

40°

94°94°

46°


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Skills Objective A

1. In nABC, // . 2. In nJKL, // .Find CF. Find JX.

In 3 and 4, use the figure at theright, in which PQ // ON, MP 5 10, and PO 5 45.

3. If MN 5 100, what is MQ?

4. If OR 5 66, what would RN haveto be so that PR is parallel to MN?

5. Use the figure below. Name allpairs of parallel lines and explain whythey are parallel. The figure is notnecessarily drawn accurately.

Uses Objective HIn 6 and 7, use the drawing of the roof below.

6. Given that TO 5 6 m, OP 5 4 m,TR 5 7.2 m, and RE 5 4.8 m, areOR and PE parallel?

7. If PE 5 12 m, what is the length ofthe support OR?

JKXYABEF


A

6

8

4A B

C

E F 1.4

1.8

4.8

X

JK

Y

L

10

45

M

PQ

O

RN

6 m

4 m

12 m

4.8 m

7.2 m

R

T

P

O

E

5

39

15

11 7

6

T

W

V

X

YU

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Vocabulary

1. If a, b, and g . 0, what proportion guaranteesthat g is the geometric mean of a and b?

Skills Objective AIn 2–4, Use n MNO below at the right.

2. If ML 5 5 and LN 5 20, then

OL 5 .

3. If ML 5 8 and OL 5 16, then

MN 5 .

4. If ON 5 15 and MN 5 17, then

LN 5 .

5. What is the length of BD in nABCat the right?

6. MK is the altitude to the hypotenuse ofn AME at the right. AE 5 10 andKE 5 9, find MA, ME, and AK.

MA ME AK

7. In the 3-4-5 right triangle TFV at the right,find the length of GF, GT, GV, and GH.

GF GT

GV GH


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M

NO

L

C

BA

D y

x

KEA

M

10

FV

T

4

3

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Skills Objective C

1. In nPQR at the right, find PQ and QR.

PQ

QR

2. In nKLM at the right, find KL and KM.

KL

KM

3. An isosceles triangle has base angles of 30°. Thelength of the base is 14. What is the length of thealtitude to the base?

4. In the diagram at the right, if JA 5 10and IE 5 12, find each length.

JI

AM

ME

JM

5. In nGSM at the right, GN 5 18 and N is themidpoint of SM. Find each length.

SN

GS

NM

GM


A

P

R

Q60°

7

K

L

M

8

I

J

M

45°

AE

N

S60°

G

M

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Skills Objective D

1. Draw a right triangle with legs measuring8 units and 15 units. Use this triangle toestimate ∠ G if tan G 5 .

2. Use nAMC at the right. Find each ratio.

a. tan M

b. tan C

Skills Objective E

3. In nJES, use a protractor to determinem∠ E to the nearest degree, and thencalculate tan E to the nearest hundredth.

Properties Objective GIn 4 and 5, use nQRS at the right. Do not measure.

4. What is the tangent of ∠ Q?

5. is the tangent of

which angle?

Uses Objective I

6. Gertie the gopher is looking at a tree that is 100 feetaway from her hole. If the angle of elevation betweenGertie’s hole and the top of the tree is 23°, how tall isthe tree?

QRRS

815


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513

M

A C

J

ES

Q

RS

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15

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Skills Objective DIn 1–3, use n JKL at the right. Find the indicated ratio.

1. sin K

2. cos K

3. cos J

Skills Objective E

4. Estimate sin 68° to the nearest hundredth.

In 5 and 6, give exact values.

5. cos 45° 6. sin 60°


7. Define cos A.

In 8 and 9, use nHIL at the right.

8. is the of angle H.

9. Write a ratio for cos H.

Uses Objective I

10. What is the measure of the angle x madeby a 200-ft supporting cable with a 150-ft-tallradio tower?

11. How far up on the side of a building can a 15-m ladder reach if the measure of the angle it makes with the ground may not exceed 72?

ILHI


A

10.5

14

17.5

KL

J

LH

I

x

Name


Skills Objective DIn 1–4, determine the area of the triangle.

1. 2.

3. 4.

Uses Objective 1

5. A cruise ship travels at 18 kilometers per hour in acourse 54° south of west. Find the southern andwestern exponents of its velocity.

6. A hiker needs to reach a point 3 miles north and12 miles east of her present location.

a. In what direction must she hike?

b. How far will she need to travel?

7. A submarine traveled 450 miles in the direction25° north of west. It then traveled 210 milesin the direction 74° north of west. To thenearest mile, how far has the submarinetraveled from its starting point?

4056°

349

7134°


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56° 5.5

83°6.8

25°106

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Skills Objective AIn 1–3, the circle has a radius of 6 cm. Findthe length of the chord of the indicated arc.

1. a 90° arc

2. a 60° arc

3. a 132° arc

In 4 and 5, use nMJK in (K with radius 3.5 in.

4. If m∠ JKM 5 84, then mCMJ 5 .

5. Find MJ and MK.

MJ MK

6. True or false. If nYES is equilateral,then m CYE 5 mCSE 5 120°.

Properties Objective FIn 7–9, square QUAD is inscribed in (C,and < bisects ∠ QCU. Justify the statement.

7. QB 5 BU

8. mCDA 5 m

CAU

9. , // QD


A

6 cm

3.5M K

J

SR

Y

E

,

D

BQ

A

U

C

Name


Skills Objective BIn 1–4, use the circle at the right. Find theindicated measure.

1. mCWXZ

2. m∠ WXZ

3. m CXZ

4. m∠ YZW

Skills Objective C

In 5–9, use (R at the right, in which BUis a diameter, m∠ U 5 37, and m CBE 5 135°.Find the indicated measure.

5. mCBL

6. m CEU

7. m∠ ELU

8. m∠ LTU

9. m∠ BTE

Properties Objective GIn 10 and 11, use ( Q at the right.

10. Explain why nRST is a right triangle.

11. If m∠ R 5 x, then mTRC5 .


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X

Z

Y

W

99°32°

46°

L

R

B

E

T

U

R

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Skills Objective DIn 1 and 2, use the right-angle method tofind the center O of the circle.

1. 2.

Uses Objective IIn 5–7, use the diagram at the right of the space shuttle. The actual shuttle is 184 feet long.

5. Locate all points where a photographer could standto fit CA exactly in the picture if the camera lens has a picture angle with measure of 65.

6. a. How far from point M, the midpoint of CA, willthe photographer have to be if she stands on theperpendicular bisector of CA?

b. Mark this point x on the diagram.

7. How far would a different photographer need to be from M if his camera lens angle measures 56 andhe stands on the perpendicular bisector of CA?

8. In general, the ? the picture angle, the greater the distance one must stand to fit an object exactly into a picture.

4. Draw a circle through thethree points C, A, and T.

3. Draw the entire circle of thecircular arc below. Thenfind the center of the circle.

BO

AR

O

Q


A

X

130°

C

A

M

25°

A

C

T

Name


Skills Objective CIn 1–3, use the circle at the right.

1. If m∠ GHI 5 52 and m CGI 5 44°, what other arcmeasure can be determined? What is its measure?

2. If m CFG 5 130° and m CIJ 5 120°,what is m∠ IHJ?

3. If m CFG 5 133° and mCIJ 5 117,°what is m ∠ GHI?

In 4–6, use (Q at the right. Assume thatmCNO 5 30° and mCRS 5 70°.

4. What is m ∠ M?

5. What is m ∠ RPS?

6. List all other angle measures that you can.


7. Write an argument to complete the proof.Given: (A with segments and

measures as given at the right.To prove: nOYC is an isosceles right triangle.


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H

J

F

G

I

P Q

S

O

M

R

N

C

KR

Y

O

x °

A

x °

165°

15°

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In 1–3, ←→GS and

←→GN are tangents to

( M at S and N.

1. a. What is the measure of ∠ GSM?

b. What kind of figure is GSMN?

2. If SM 5 8 and GS 5 15, find the area of GSMN.

3. If m∠ SMN 5 174, what is m CSE?

4. Write an argument to complete the proof.

Given: m∠ OPQ 5 90; OP // ,.To prove: , is tangent to (P at point Q.

Uses Objective JIn 5–7, assume the radius of the earth is3960 miles or 6375 kilometers and that thereare no hills or obstructions.

5. If a person could stand on the tip of the torch of theStatue of Liberty, 91.5 m above the water, how farcould the person see?

6. The most-recently accepted height of Mt. Everest is29,028 ft. How far could a person see when standing on its summit?

7. To the nearest foot, how far above the earth is a planeif the pilot can see 8 miles?


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Skills Objective CIn 1–4, use the figure at the right, with P the center of the circle. Find the indicated measure.

1. m∠ JIK

2. mONC

3. mLMC

4. m∠ JIM

In 5–7, use (P at the right. Find theindicated angle measure if mTAC 5 35°.

5. m∠ LSE

6. m∠ SLE

7. m∠ SAL


8. Given that XY and XZ are tangent to (U at points Y and Z, if m

CYWZ = 236° list all of the

angle measures and arc measures that you can.



Given:←→WV is tangent to (T at V,

and ←→WV // SU.

To prove: nSVU is isosceles.


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Skills Objective E

1. L, M, N, and O all lie on (Q at the right. IfLP 5 33, PN 5 2, and MP 5 3, find OP.

In 2 and 3, use (R at the right.

2. If EA 5 18 and AL 5 2, find LH.

3. If LH 5 10 and EA 5 25, find AL.

In 4 and 5, use (M at the right.

4. If AK 5 4, KS 5 10, and AN 5 5, find PN.

5. If AS 5 16, AK 5 6, and AP 5 12,

a. find AN.

b. find PN.

6. Use (C at the right. ←→AB is tangent

to (C at B.

a. What is the power of point A?

b. If AJ 5 18 and JK 5 60, find AB.


A

QL

MN

PO

R

E

A

L H

S

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A

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M

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B J

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1. Consider all figures with area 40 square centimeters.

a. Which has the least perimeter?

b. What is that perimeter?

2. A circle and a regular polygon each have a perimeterof 10 inches. Which has less area?

3. Of all triangles with a fixed perimeter, which has the greatest area?

4. A rectangle has sides measuring 2.5 mm and 3.8 mm.What is the greatest possible area of a figure withthe same perimeter?

Uses Objective K

5. a. Draw a figure with a large b. Draw a figure with a largearea for its perimeter. perimeter for its area.

6. Northfield wants to build a park with an area of about8000 square meters.

a. What would be the least perimeter possiblefor the park?

b. If a park with a quadrilateral shape is desired, whatwould be the least perimeter?

7. A fence encloses a grazing area as shownat the right.

a. Find the area of the region.

b. Find the area of the largest region thatcould be enclosed by this fence.


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200 ft

250 ft

500 ft 570 ft

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1. Consider all figures in space with a surface areaof 100 square feet.

a. Which has the greatest volume?

b. What is this volume?

2. A sphere and a square-based pyramid each havea volume of 2400 cm3. Which has the greatersurface area?

3. Of all regular pyramids or cones with equalsurface area, which has the greatest volume?

4. Of all the rectangular prisms with a fixedvolume, which has the least surface area?

Uses Objective K

5. a. Draw an object with a large b. Draw an object with a largevolume for its surface area. surface area for its volume.

6. A container is to be designed for 36 in3 of butter. Find its surface area if the container is

a. a cube.

b. a sphere.

c. a right cone with height 4 inches.

7. Give two reasons why the cube would be the bestcontainer in Question 6.

8. A jeweler has enough gold to cover a surface area of 3 cm2

with a particular thickness of gold.

a. What is the volume of the largest bead she could cover?

b. What is the volume of the largest prism-shaped beadshe could cover?

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Vocabulary

1. What is a pixel? Samples are given.

2. How is the resolution of a computer screen measured?

Skills Objective AIn 3–6, draw the line or lines described.

3. an oblique discrete line 4. a vertical discrete line

5. a vertical discrete line intersecting 6. two oblique discrete lines thata horizontal discrete line at a point cross but have no point in common


7. In discrete geometry, a point has no size. ____________________

8. In discrete geometry, two lines mayintersect at two distinct points. ____________________

Uses Objective J

9. What is represented by the discreteline in the road-atlas legend atthe right?

______________________________


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A pixel is a tiny dot which makes updisplays on TV screens, computermonitors, and graphics calculator windows.

It is measured in pixels per square inch.

false

false

Scenic Route

Toll Limited-Access HighwayPrincipal HighwayOther RoadUnpaved RoadScenic Route

Samples are given.

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1. True or false. A point as a location has thickness.

2. Describe a property that holds for lines in syntheticgeometry but not for lines in discrete geometry.

3. Give a possible coordinate forpoint B on the number line atthe right.

Uses Objective I

4. At the Mt. Washington, NH, weather station the greatestwind speed on record is 231 mph, while the average windspeed is 35.3 mph. The greatest wind speed is how much more than the average?

5. A tailor had her customers stand on a 9 0-high platform to mark hems with a yardstick. If she marked a hem at 31 0from the floor, how far would the hem actually beoff the ground?

6. According to the 1995 Information Please Almanac, the record high temperature in Antarctica was 15˚C in 1974,and the record low temperature was -89˚ C in 1983. Howmuch colder was the record low than the record high?


In 7–8, use the number line atthe right.

7. Find the distance from a. A to C. ________ b. C to B. ________

8. If F is on this number line, and AF 5 12,give the two possible coordinates of F. _________ _________

34

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A

false

Sample: In synthetic geometry, lines are

dense.

Sample:100.3

195.7mph

22 ''

1048

14

11 7

10 -14

0 3

A CB

100 101

A CB

Name



1. True or false. In coordinate geometry, lines are dense. _________________

2. Name two number lines in theCartesian plane. __________ __________

Representations Objective LIn 3–6, give two points on the line with thegiven equation. Then graph the line.

3. y 5 -4x 1 6 4. x 1 2y 5 8

5. y 5 -3 6. x 5 4

In 7–9, use the equation to classify the line asvertical, horizontal, or oblique.

7. y 5 -462 8. 7x 1 y 5 1 9. x 5 1259

horizontal oblique vertical_______________ _______________ _______________

x y

4 04 1

x y

1 -32 -3

x y

0 42 3

x y

0 61 2


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x-axis y-axis

5

-5

5-5

y

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-5

5-5

y

x

5

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5-5

y

x

5

-5

5-5

y

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Sample points are given.

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Skills Objective B

1. Consider the network at the right.

2a. How many odd nodes are there? ________

7b. How many even nodes are there? ________

c. Is the network traversable? If it is, give a path.

Yes; sample: FABCDEADBEFGHIC

2. Draw a network consisting of five lines and three nodes.

Sample:


3. In graph theory, there may be four distinct lines betweentwo points. true

4. In graph theory, a point has size and shape. false

5. Multiple choice. A line is dense in which geometry? b(a) discrete geometry

(b) plane coordinate geometry

(c) graph theory


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5

Uses Objective J

6. Use the street plan at the right.

a. Model the street plan as a network of lines and nodes.

b. Would a mail carrier be able to plan a route to visit all the houses on the streets without retracing his steps? Why or why not?

Yes; there are exactly 2 odd nodes.

7. At the right is a diagram of San FranciscoBay and the counties bordering it. The bayis crossed by 7 toll bridges.

a. Represent the bridges and land massesas a network of lines and nodes.

b. Is the network traversable? Yes

c. In 1989, the Bay Bridge, which connects San FranciscoCounty with Alameda County, was closed after an earthquake.Without the use of the Bay Bridge, is the network traversable?Justify your answer.

Yes; there are exactly 2 odd nodes.

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Sonoma Co.

San Francisco Co.

Solano Co.Marin Co.

Contra Costa Co.

San Mateo Co.

Alameda Co.

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Skills Objective C

In 1–2, tell whether the drawing is a perspective drawing.

no yes1. __________ 2. __________

3. Change the drawing to a perspective drawing.

Sample:

In 4–6, locate the vanishing point of the drawing.

4. 5. 6.


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milk carton

highway buildingtable tennis

football field

fence

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Properties Objective EIn 1 and 2, tell how many dimensions the object has.Ignore small thicknesses.

3 21. a shoebox ____________ 2. a paper clip ____________

Properties Objective GIn 3–6, in what type of geometry could the triangle appear?

3. 4.

graph theory plane coordinate____________________________ ____________________________

5. 6. geometry

synthetic geometry discrete geometry____________________________ ____________________________

Samples are given.In 7–9, true or false. If false, rewrite to make it true.

7. Space is an undefined term in geometry. false; space is defined as the set of all points in geometry.

8. A line looks the same in any geometry. false; a line looksdifferent in different geometries.

9. Coordinate geometry is the study offigures in three dimensions. false; coordinate geom-etry is the study of figures in two dimensions.

In 10–12, multiple choice. Choose the geometry in which the statement is true.

10. A line is dense. b(a) graph theory (b) synthetic geometry (c) discrete geometry

11. More than one distinct line may contain two given points. a(a) graph theory (b) plane coordinate geometry (c) synthetic geometry

12. Points have a definite shape. b(a) synthetic geometry (b) discrete geometry (c) plane coordinate geometry


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1. Which two geometries are Euclidean?

synthetic geometry plane coordinate______________________________ ______________________________

2. What is the difference between a postulate and a theorem?

Sample: A postulate is an assumption; a

theorem is a statement that follows from definitions, postulates, or previously provedtheorems.

In 3–6, tell which assumption from the Point-Line-Plane Postulate permits the procedure described.

Unique Line Assumption Number Line Assumption


3. Locating point P not on line m.


4. Graphing the line with equation y 5 2x byconnecting the ordered pairs (1, 2) and (2, 4).

Unique Line Assumption

5. Locating point P not on plane m.


6. Measuring the length of a segment by holdinga ruler to it.

Number Line Assumption

In 7–12, tell if the statement is true or false in Euclidean Geometry.

7. A point can be represented by an ordered pair of real numbers. true

8. Lines have no thickness. true

9. Any line may be coordinatized using the real numbers. true

10. Two different lines cannot intersect at two different points. true

11. Lines are not dense. false

12. Lines are parallel if they have no points in common or ifthey are the same line. true


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Skills Objective D

In 1–4, use←→LP at the right.

1. Name↔LP in two other ways. 2. Name

→ON in two other ways.

Samples: ←→LM,

←→MN,

→OM,

→OL_____________________________ __________________←→

MP, ←→LO,

←→PL

false3. True or false. MN 5 MN ____________→ML4. Name the ray opposite to

→MN. ____________

5. Draw →TS with M between T and S.

6. Draw AB with S between A and B, and M between S and B.

Properties Objective HIn 7 and 8, use the figure at the right. A isbetween M and P.

277. If MA 5 17.3 and AP 5 9.7, what is MP? ________

568. If AM 5 36 and PM 5 92, what is AP? ________

9. On a number line O is between X and Y. If X hascoordinate -18 and Y has coordinate -2, give therange of coordinates for O. -18 < O < -2

10. On a number line, P is between L and M. If L hascoordinate -20, M has coordinate 3, and LP 5 5,what is the coordinate of P? -15

In 11 and 12, use the number lineat the right. Write an inequality todescribe the coordinates of points on

x ≥ -20 -40 ≤ x ≤ 2011.→BD. _______________________ 12. AC. _______________________


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P

M

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A B C D

-40 -20 20 400

Sample: T M S

Sample: A MS B

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Skills Objective AIn 1–6, tell whether the set is convex or nonconvex.

1. 2. 3.

nonconvex nonconvex convex________________ ________________ ________________

4. 5. 6.

convex nonconvex convex________________ ________________ ________________

In 7 and 8, draw the figure. Samples are given.7. a convex 10-sided region 8. a nonconvex 6-sided region

9. Draw a segment to show that the set atthe right is not convex.

10. Imagine that a friend cannot understand why Figure 1 below is aa convex set, but Figure 2 is not convex. Write an explanation to help your friend with the problem.

Fig. 1 Fig. 2

Sample: In Fig. 1, the interior of the angle is included, while the interior is not includedin Figure 2. A connecting segment in Fig. 1 is contained in the set, while that is not the case in Fig. 2.


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Skills Objective CIn 1–3, let r 5 the cake falls; s 5 you stamp your foot; and t 5 the floor shakes. Write the conditional in words.

If the floor shakes, then the cake falls.1. t ⇒ r

If you stamp your foot, then the floor shakes.2. s ⇒ t

If you stamp your foot, then the cake falls.3. s ⇒ r

4. Write in symbols: If a, then b. a ⇒ b

Properties Objective GIn 5 and 6, underline the antecedent once and the consequent twice.

5. If wishes were horses, then beggars would ride.

6. Make yourself a sandwich if you’re hungry.

In 7 and 8, rewrite as a conditional. Samples are given.7. All 21-year-olds are eligible to vote.

If a person is 21 years old, then theperson is eligible to vote.

8. Every sign must be written in English and in Spanish.

If a sign is written, then it must be written in English and in Spanish.


9. Consider the BASIC 10 INPUT Qprogram at the right. 20 IF Q >= 15 THEN PRINT “GOODBYE”

30 END

a. Give a value for Q which will cause the computerto print “GOODBYE”. Sample: 21

b. Give a value for Q which will cause the computer to printnothing. Explain your answer using a conditional statement.

Sample: 10; if the value of Q is less than15, the computer will not execute line 20.

11 ©

10. True or false. An instance of “if w, then z” is a situationin which w is true and z is false. false

In 11 and 12, a conjecture is given. Determinewhether each example is

ii(i) an instance of the conjecture.

i(ii) a counterexample to the conjecture.

(iii) neither an instance nor a counterexample to the conjecture.

11. If t ≥ 40, then t ≥ 41.

a. t 5 45 b. t 5 39 c. t 5 40.3

i iii ii________________ ________________ ________________

12. If a figure has four sides, then its interior is a convex set.

a. b. c.

iii i ii________________ ________________ ________________

Uses Objective K

13. An ad said, “If you buy a refrigerator before Friday, you’llreceive a $100 rebate.” Danielle bought a refrigerator onThursday of the same week. What will happen?

Danielle will receive a $100 rebate.

14. If the service at a restaurant is very good, then Lonnie willleave a 20% tip. Today, Lonnie thought the service was verygood. What will happen?

Lonnie will leave a 20% tip.

15. Rewrite as a conditional: It is always cloudy when it rains.

If it rains, then it is cloudy.

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Skills Objective C

1. Write the converse of c ⇒ d. d ⇒ c

Properties Objective EIn 2 and 3, true or false.

2. The converse of a true statement is always true. false

3. The converse of a false statement is always false. false

In 4 and 5, a conditional is given. a. Is the conditionaltrue? If not, give a counterexample. b. Write its converse.

Samplec. Is the converse true? If not, give a counterexample. counter-

examples are given.4. If the light is red, the traffic stops.

a. No; a policeman could direct traffic.

b. If the traffic stops, the light is red.

c. No; traffic could stop for an accident.

5. If m > 0, then m ≥ 2.

a. No; m 5 1.5

b. If m ≥ 2, then m > 0.

c. Yes

6. Use the consequent You can vote in the U.S. presidential election.Write a conditional which is true but whose converse is not true.

Sample: If you are 21 years old, then you can vote in the U.S. presidential election.

Uses Objective K

7. An ad said: “If you use our shampoo, your hair will be clean andshiny.” Jessica read the ad and wanted clean and shiny hair. A month later her hair was very clean and quite shiny. Did Jessica usethe shampoo? Write an explanation for your answer in which you use the converse of the conditional in the ad.

Sample: The ad’s converse is: “If your hair is clean and shiny, then you used our shampoo.” But, we do not know if Jessica used the shampoo.


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Skills Objective C

1. Let p 5 water freezes and q 5 the temperature is lessthan or equal to 08 C. Write p ⇔ q in words.

Water freezes if and only if the temperature is less than or equal to 08 C.

Skills Objective DIn 2 and 3, use the number line atthe right.

2. Find the coordinate of each point.

a. the midpoint of AB -100

b. the midpoint of BC 150

c. the midpoint of AC -50

3. If A is the midpoint of CD, what is the coordinate of D? -800

4. M is the midpoint of GH,GM 5 2(x 1 5), andMH 5 10x 2 13.Find GH.

, or 31.5

5. X is equidistant from Z and W,XW 5 9 2 2t, and ZW 5 6t 2 9.Find ZX.

, or 3.6


6. Consider this definition of tangent: A tangent to a circle is a linewhich intersects the circle in exactly one point. Name two undefinedgeometric terms used in this definition. Samples are given.

line point__________________________ __________________________

Z WX 118855

G

H

M 663322

A

-300 -200 -100 0 100 200 300

B C


A

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7. Consider the definition for a chord of a circle: A segment is achord of a circle ⇔ its endpoints lie on the circle.

a. Write this definition as two conditionals.

If a segment is a chord of a circle, then its endpoints lie on the circle.If the end points of a segment lie on acircle, then the segment is a chord of

the circle.

b. Which conditional goes in the direction characteristics ⇒ term?

If the endpoints of a segment lie on a circle, then the segment is a chord of

the circle.

In 8–10, tell which property of a good definition isviolated by the “bad” definition. Samples are given.8. A polygon is a figure with sides and angles.

The definition is not accurate.

9. A line segment is a straight path between two points.

The definition uses undefined terms.

10. A line segment consists of two points and all the pointsbetween them, and it is always straight and can be measuredto find its length.

The statement contains too much information.

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Vocabulary

1. What is the meaning of each symbol?

intersection uniona. > ________________________ b. < ________________________

empty setc. { } ________________________

Properties Objective I

2. Let M 5 {100, 200, 300, 400, 500} and N 5 {0, 1, 2, 3, . . ., 200}.Identify the set.

{100, 200} {0, 1, 2, . . ., a. M > N _____________________ b. M < N _____________________

199, 200, 300, 400, 500}3. Let R 5 set of real numbers s with s ≥ 15 and

P 5 set of real numbers s with s ≤ 20.

a. Draw R and P on thenumber line at the right.

b. Write an inequality for R > P. {s: 15 ≤ s ≤ 20}

c. Describe R < P. all real numbers

4. Use the figure at the right. Give

a. the segment(s) of DFER > DREU. ERb. the segment(s) of DFER < DREU. FE, EU,

UR, RF., ERc. the segment(s) of quadrilateral FIGU > DFRU. FUd. the segment(s) of quadrilateral FIGU > DERU. EUe. the point(s) of IG > ER. { }

f. the segment(s) of pentagon FIGUR < DFRE. FI , IG, GU,UR,

5. Let G 5 the set of all states east of the Mississippi RF, FE,ERRiver and H 5 the set of all states west of the Mississippi River. Describe each set.

all states in U.S. { }a. G < H _______________________ b. G > H _____________________

6. Draw→PT and

→RT 7. Draw

←→MN and AB

so that→PT >

→RT 5 T. so that

←→MN > AB 5 AB.

Sample: Sample:

M B A NP RT

I

U

R

F

G

E

-10 -5 0 5 10 15 20 25P

R

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1. Draw a nonconvex decagon. 2. Draw a convex pentagon.

Sample: Sample:

3. Use the definition of polygon to explain why the figureat the right is not a polygon.

Sample: Some segments intersect morethan two others rather than exactly two.

4. Draw a figure that is the union of four segments but is not a quadrilateral. Sample:

Uses Objective L

5. Some tables for playing games are shaped like the polygonat the right.

a. Name the polygon. hexagonb. Give a reason for its shape.

Sample: One player can sit at each edge.

6. The playing area at Oriole Park at Camden Yards inBaltimore, MD, is diagrammed at the right.

a. What name is given to the large polygon?

hexagonb. The small figure is not a polygon. Why?

Sample: Four sides are not segments

Representations Objective N7. Draw a hierarchy relating the following

terms: figure, polygon, triangle, quadrilateral, isosceles triangle, nonagon.

17

figure

polygon

triangle quadrilateral nonagon

isosceles triangle

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Properties Objective JIn 1–5, can the numbers be lengths of the sides ofa triangle? Justify your answer.

1. 10, 15, 25 No; 10 1 15 5 25

2. 4, 4, 4 Yes; 4 1 4 > 4

3. 4, 15, 11 No; 4 1 11 5 15

4. , , Yes; 1 > , 1 > , 1 >

5. 1, 10, 10 Yes; 1 1 10 > 10, 10 1 10 > 1

6. Stu drew and labeled the triangle below. Why didhis teacher give him no credit for his drawing?

Sample: The sum of two side lengths must be greater than the length of the third side, but

AB 1 AC 5 16 1 18 < BC.7. Two sides of a triangle measure 92 meters and

54 meters. How long can the other side be? 38 m < x < 146 m

8. In DISO at the right, if IS 5 30 how long can SO be?

O < SO < 60

Uses Objective M

9. It is a 35-minute drive from Kyoko’s house to Anne’s house and a 25-minute drive from Anne’s house to Ben’s house.Using just this information, tell how much time it wouldtake to drive from Kyoko’s house to Ben’s house. between

10 min and 60 min10. It is 44 miles from Lancaster, Pennsylvania, to Harrisburg,

Pennsylvania, and 78 miles from Harrisburg to Reading,Pennsylvania. Using just this information, tell how farit is from Reading to Lancaster. between

34 mi and 122 mi

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C

A

B

16

36

18

1133

1166

1144

1166

1133

1144

1144

1133

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1. A conjecture is always true. false

2. One counterexample will show that a conjecture is false. true

3. People make conjectures based on patterns they observe. true

In 4–7, a conjecture is given. Determine whethereach example is an instance of the conjecture, acounterexample to the conjecture, or neither aninstance nor a counterexample of the conjecture.

4. If x is a positive number, x 2 ≥ 1.

a. x 5 5 b. x 5 -2 c. x 5 .5

instance neither counter-________________ ________________ ________________

example5. If x ≤ 0, then x ≤ -1.

a. x 5 10 b. x 5 - c. x 5 -3

neither counter- instance________________ ________________ ________________

example6. If the area of rectangle RECT is 100, then the perimeter of RECT is 40.

a. b. c.

instance neither counter-________________ ________________ ________________

example7. If XT 5 TY, then T is the midpoint of XY.

a. b. c.

neither counter- instance________________ ________________ ________________

example

X

Y

T5.0

5.0

X

YT

5.0

5.0

X

Y

T5.1

4.9

C

E

T

R4

20

C

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T

R

10

10

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Skills Objective A

1. Draw and label an ∠ CDB 2. Draw and label an ∠ XTL with

with measure 124. measure 86 and its bisector→TA.

Sample:

In 3–7, use the figure at the right.

3. Give two other names for ∠ APT.

∠ TPA ∠ 4___________ ___________

∠ TPL ∠ APE4. Name two distinct straight angles. ___________ ___________

5. True or false. ∠ LTP has measure 0. true

6. m∠ APT 1 m∠ TPR 5 m∠ APR is an exampleof the ....?.... Property. Angle Addition

7. If m∠ LPR 5 127, find each measure.

37 53a. m∠ LPE ____________ b. m∠ TPR ____________

37 143c. m∠ TPA ____________ d. m∠ APL ____________

In 8–10, true or false. Use the diagramat the right. You can assume from thediagram that

8. X, W, and Z are collinear.

true_______________

false9. →WY bisects ∠ QWZ.

true10. Y is in the interior of ∠ QWZ.


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Skills Objective B

11. Suppose m∠ ATC 5 145, m∠ ATY 5 6b 1 10, and m∠ CTY 5 3b 1 9.

a. Find b. 14

b. Find m∠ ATY. 94

12. Suppose →IT bisects m∠ BIS,

m∠ BIT 5 12x 1 3, and m∠ TIS5 10x 1 10.

Find m∠ BIS. 90

13. Let m∠ POR 5 5t, m∠ ROL 5 3t 1 8, and m∠ POL 5 9t 2 1.

a. Write an equation to find t.

5t 1 3t 1 8 5 9t 2 1

b. Find t and each measure.

9 45 35 80t ______ m∠ POR ______ m∠ ROL ______ m∠ POL ______

c. Does→OR bisect ∠ POL? no

14. Suppose in the figure for Question 13,→OR bisects ∠ POL,

m∠ POR 5 5t, and m∠ POL 5 9t 1 10.

a. Write an equation involving t. b. Find t.

2(5t) 5 9t 1 10 t 5 10________________________ ______________


15. What property did you use to writethe equation in Question 13a? Angle Addition

16. What justification allows you to writethe equation in Question 14a? angle-bisector

definition

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Skills Objective E

1. Rotate AB 658 about O. 2. Rotate DPQR -908 about M.

3. Draw the image of MNOP after it 4. B is the image of A after ahas been rotated -908 about O. rotation about point O. What

is the magnitude of the rotation?

-1208

In 5 and 6, true or false.

5. A rotation of 908 is equivalent to a rotation of -908. false

6. A rotation of 1808 is equivalent to a rotation of -1808. true

Skills Objective F

7. Use (O at the right, in which mXBC 5 33° andm∠ COD 5 90. Find each measure.

a. m∠ BOC 33

b. mXAB 1478

c. mXBCD 1238

d. mXBAC 3278


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8. Use (P at the right, in which PT. bisects ∠ SPO and m∠ SPD 5 50. Find each measure.

a. mXSDO 2308

b. mXDT 1158

c. mXTOD 2458

Uses Objective I

9. A pendulum swings from point A topoint B and back. Measure to find themagnitude of the rotation

a. from B to A. -808

b. from A to B. 808

10. A circular cake is to be cut into 10 wedgesof the same shape and size.

a. What is the degree measureof the arc of each wedge? 368

b. If two people don’t wantcake and the cake is insteaddivided equally among the rest of the guests, what is the degree measure of the arc of one wedge? 458

11. The dial on a clothes dryer can beset from Hot to Air and to four settings in between. Arc measures from any position to the next are the same. If the dial is turned from 1 to 4, what is the magnitude of the rotation? -1088

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Skills Objective AIn 1–3, use the diagram at the right.

1. Name two linear pairs. Samples are given.∠ BTO, ∠ OTS ∠ OTA, ∠ ATR___________________ ___________________

2. Name a pair of vertical angles.

∠ BTO, ∠ STR___________________

3. If m∠ OTB 5 45, find the measure of as many otherangles as possible.

m∠ BTR 5 135, m∠ RTS 5 45, m∠ OTS 5 135

684. a. If m∠ 8 5 112, then m∠ 9 5 _________

112and m∠ 10 5 _________ .

180 2 10kb. If m∠ 8 5 10k, m∠ 11 5 _______________

10kand m∠ 10 5 _________ .

Skills Objective B

5. Suppose m∠ AOB 5 10x 2 6 and m∠ DOC 5 14x 2 20.Find each measure.

29 151a. m∠ DOC _________ b. m∠ BOD _________

6. Suppose m∠ 1 5 11n 1 13 and m∠ 4 5 5n 2 9.

134a. Find m∠ 3. ________________

b. Explain how you found the answer in Part a.

Sample: ∠ 1 and ∠ 4 are a linear pair, so they are supplementary and m∠ 1 1 m∠ 4 5 180. Then 11n 1 13 1 5n 2 9 5 180, n 5 11, and m∠ 1 5 134. Since ∠ 1 and ∠ 3 are vertical

angles, m∠ 1 5 m∠ 3 5 134.


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6. Two angles are complementary. The measure of thelarger is 10 more than 4 times the measure of thesmaller. Find the measure of each angle. 16 74_________ _________

7. Students were asked to draw two non-adjacent supplementaryangles with the one angle 4 times as large as the other.

Art drew this diagram. Augie drew this diagram.

Neither student received full credit. Explain why and drawa correct answer.

Art drew adjacent supplementary Sample:angles; Augie drew non-adjacentcomplementary angles.

Uses Objective I

8. Draw the path of a plane flying 758 9. A plane’s course is 608 east ofsouth of west. north. How far must it turn to

change its course to due east?

-308______________

10. A mirror which is hinged to a wall is moved so thatit makes an 838 angle with the wall. What is themeasure of the other angle it makes with the wall? 97

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Properties Objective GIn 1–5, multiple choice. Each statement illustrates one of theproperties listed at the right. Give the letter of the property.

1. If 1.7g2 5 22, then 22 5 1.7g2. b

2. If 3x 1 5 5 -2, then 3x 5 -7. f

3. m∠ ABC 5 m∠ CBA. a

4. If GK 5 MN and MN 5 RQ, cthen GK 5 RQ.

5. If t 5 9, then t 5 27. g


6. If AB 1 BL 5 19 and BL 5 LE, whatcan you conclude about AB and LE by theSubstitution Property?

AB 1 LE 5 19

7. If AB 5 BL and LE 5 AB, what canyou conclude by the Transitive Property?

BL 5 LE8. a. Write an equation relating ∠ PQR, ∠ PQS,

and ∠ RQS.

m∠ PQS 1 m∠ RQS 5 m∠ PQRb. Use the Equation to Inequality Property to write a

true statement involving ∠ SQR and ∠ PQR.

m∠ PQR > m∠ SQRc. Is it possible to conclude that m∠ SQR < ∠ PQS?

Give a reason for your answer.

No; measures of anglescannot be assumed from a diagram.

9. Suppose m∠ T 5 90 2 x. Find m∠ T if

41 90 2 5aa. x 5 49. ____________ b. x 5 5a. ____________

c. What property did you use in Parts a and b? Substitution

13


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(a) Reflexive Property of Equality

(b) Symmetric Property of Equality

(c) Transitive Property of Equality

(d) Equation to Inequality Property

(e) Substitution Property

(f) Addition Property of Equality

(g) Multiplication Property ofEquality

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Properties Objective HIn 1–3, multiple choice. Choose the correct justificationfor the given conclusion.

1. Given: m∠ POT 5 109. cConclusion: ∠ POT is obtuse.

(a) Angle Addition Postulate

(b) Vertical Angles Theorem

(c) definition of obtuse angle

2. Given: I is the midpoint of MN. bConclusion: MI 5 IN

(a) Additive Property for Segments

(b) definition of midpoint

(c) Equation to Inequality Property

3. Given: ∠ 4 and ∠ 3 are a linear pair. aConclusion: ∠ 3 and ∠ 4 are supplementary.

(a) definition of supplementary angles

(b) Linear Pair Theorem

(c) Angle Addition Postulate

In 4–6, write a justification for each conclusion.

4. Given: MA 5 AN.Conclusion: DMAN is isosceles.

definition of isoscelestriangle

5. Given ∠ POT and ∠ ROE are vertical angles.Conclusion: m∠ POT5 m∠ ROE.

Vertical Angles Theorem

6. Given: (P.Conclusion: T and L are equidistant from P.

definition of circle


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Skills Objective AIn 1–4, use the figure at the right inwhich m // <.

1. Name the transversal of m and ,.

line s2. Name two pairs of corresponding angles.

∠ 1, ∠ 3 ∠ 2, ∠ 4__________________ __________________

3. If m∠ 2 5 117, find the measure of each angle.

63 63 117 117 63∠ 1 _______ ∠ 3 _______ ∠ 4 _______ ∠ 5 _______ ∠ 6 _______

4. If m∠ 1 5 4e, find the measure of each angle.

18024e 4e 18024e 18024e 4e∠ 2 _______ ∠ 3 _______ ∠ 4 _______ ∠ 5 _______ ∠ 6 _______

5. Use the figure at the right in which p // q.If m∠ 1 5 5j 2 27 and m∠ 2 5 3j 1 5, findthe measure of each angle.

53 53 127∠ 1 _______ ∠ 2 _______ ∠ 3 _______

127 53∠ 4 _______ ∠ 5 _______

Properties Objective HIn 6–8, in the figure at the right, < // m. Justifythe conclusion.

6. m∠ 2 5 m∠ 8.

Vertical Angles Theorem

7. m∠ 1 5 m∠ 3.

Corres. Angles Post.

8. ∠ 3 and ∠ 6 are supplementary.

Linear Pair Theorem


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In 9 and 10, name the theorem which justifies the conclusion.

9. Line r is parallel to line i. Line i is parallel to line t.Line t is parallel to line e. Therefore, line r is parallel to line e.

Transitivity of Parallelism

10. Line r has slope - and line t has slope - .Therefore, line r is parallel to line t.

Parallel Lines and Slopes


11. Give the slope of the line passingthrough (-4, 1) and (-5, -3). 4

12. If c ≠ a, give the slope of the linethrough (a, b) and (c, d ).

In 13–15, give the slope of each line inthe figure at the right.

13.←→MP 0

14.←→PQ

-

15.←→MQ undefined

16. Find the slope of each line.

-5a. y 5 - x 1 7 ____________ b. -5x 2 y 5 10 ____________


17. What is the slope of any line parallel toa line with slope -1.3? -1.3

18. What is the slope of any line parallel to←→PQ in Questions 13–15?

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Skills Objective C

1. In the figure at the right,←→EF ⊥

←→FG

and m∠ EFG 5 3x 2 7.

, or 32Find x. ____________

2. Given: s ⊥ r; m∠ 1 5 12w 2 6.

a. Find w. w 5 8

b. If m∠ 2 5 14w 2 22,what is m∠ 2? 90

c. Is t ⊥ r? yes d. Is s // t? yes____________ ____________

Properties Objective HIn 3–6, use quadrilateral MNOPat the right.

Given: MP←→

// ←→QR,

←→QR ⊥

←→MN,

←→QR ⊥ PR

←→, and

←→NO //

←→QR.

Multiple choice. Choose the correctjustification for the conclusion.

d______ 3.←→MP //

←→NO.

c______ 4. ∠ RQN is a right angle.

b______ 5.←→MP ⊥

←→PR.

a______ 6.←→MN //

←→PO.


7. Find the slope of any line perpendicular to a line

a. with slope . _________ b. with equation y 5 -8x 1 5. _________

-2c. with equation -2x 1 4y 5 9. ____________

8. a. Find the slope of ←→PQ. ____________

b. Give the slope of any line ⊥ to ←→PQ. ____________

c. Draw the line ⊥ to ←→PQ through the origin.

4455

(a) Two Perpendiculars Theorem

(b) Perpendicular to ParallelsTheorems

(c) definition of perpendicular line

(d) Transitivity of ParallelismTheorem

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Skills Objective DIn 1 and 2, construct the perpendicular bisectorof the given segment.

1. 2.


a. Draw the line through Q ⊥ to m.Label the intersection R.

b. Draw the line through Q // to m.

Label the line ←→QT.

c. How are ←→QT and

←→QR related?

←→QT ⊥

←→QR.

Uses Objective J

4. The home at the right needs a gas pipeconnected to the gas main under the street.If the furnace is located in the southwestcorner of the home, draw the shortestpipe from the furnace to the gas main. Themain electrical line runs parallel to the gasmain. Draw the shortest line from theelectrical box to the main electrical line.

5. The lines drawn in Question 4 are

parallel_____________________________ .

6. A park district built a foot path from theparking lot to the pond at the right. Theywish to add a parallel road for trucks fromthe parking lot to the end of the pond.Draw in the road for the trucks.

7. A path is to be made from the truck roadto the bait shop. Draw the shortest path.


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STREETGAS MAIN

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PARKING LOT

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Skills Objective A

1. a. Draw rs(P).

b. Draw rs(T).


B Ba. rp(A) 5 _______ b. rs(C) 5 _______

E Ac. rt(D) 5 _______ d. rs(E) 5 _______

3. Draw the reflecting line 4. Can r,(T) 5 T'? Give a reason forp so that rp(M) 5 N. your answer.

No; TT' is not perpendicular to <.


5. If rs(P) 5 P', then s is the perpendicularbisector of PP'. true

6. If P is on ,, then r, (P) does not exist. false

7. If r←→MN

(P) 5 P', then PP' bisects ←→MN. false

8. A reflection is a transformation thatmaps a preimage onto an image. true

Representations Objective KIn 9–12, give the coordinates of the image.

(-7, -4) (-2, 1)9. rx-axis(-7, 4) ____________ 10. ry-axis(2, 1) ____________

(-6, 0) (m, -n)11. rx-axis(-6, 0) ____________ 12. rx-axis(m, n) ____________


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1. Draw r,(PQ). 2. Draw rm(TOMN). 3. Draw r←→AB

(G).

4. Draw the reflecting line such that figure II 5. Draw and label a figure for thisis the image of figure I over that line. condition r←→

AB(SAME) 5 PANT

Sample:

Properties Objective E6. True or false. Reflections preserve angle measure,

distance, and orientation. false

7. In the figure at the right, r←→OT

(MOT) 5 SOT.

38a. If m∠ MOT 5 19, then m∠ MOS 5 _______ .

90b. m∠ MXO 5 _______ .

SX 5.050c. If MS 5 10.10, then MX 5 _______ 5 _______ .

120d. If MT 5 120, then TS 5 _______ .


8. a. Draw ry-axis(PQRS).

b. List the coordinates of the vertices ofrx-axis(PQRS).

P’ 5 (-4, 2) Q’ 5 (5, -1)____________ ____________

R’ 5 (0, -3) S’ 5 (-3, -2)____________ ____________


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Uses Objective I

1. Draw the path to bounce ball B off the wall andhit ball A. Mark any congruent angles made bythis path.


a. Draw a path from the tee to the hole with abounce off wall x.

b. Draw a path from the tee to the hole with abounce off wall y.

c. Is there a direct path from the tee to thehole?

no________


a. Draw a path from the tee to the hole using abounce off one wall.

b. Draw a path from the tee to the hole usingbounces off two walls.

c. Which path would you prefer to use? Tellwhy.

The first path; it offers

less chance of error.

4. Use the billiard table at the right.

a. Draw a path to bounce ball Aoff wall p and hit ball B.

b. Draw another path from ball A to ball B with one bounce off a wall.

c. Draw a path to bounce ball Aoff wall m and then offwall n, and finally hit ball B.

d. Draw a path to bounce ball Aoff wall p and then off wall o,and finally hit ball B.

e. If the actual billiard table, withadditional balls in place, lookedlike the diagram at the right, whichpath from Parts a–d would be best?

the path in Part c


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Skills Objective DIn 1 and 2, draw the indicated reflection image.

1. rm 8 r,(PQRS) 2. rq 8 rp(ABCD)

Properties Objectives F and GIn 3–5, true or false.

3. Translations do not preserve orientation. false4. If rp 8 rq(Figure N) 5 Figure M and p // q, true

then M is a translation image of N.

5. Under a composite of reflections over two parallel lines false10 inches apart, a point is 5 inches from its image.

6. In the figure at the right, t // s, rt(ABCD) 5A'B'C'D, and rs(A'B'C'D) 5 A''B''C''D''.

1 cma. If AA'' 5 1 cm, then CC'' 5 __________ .

b. If AA'' 5 1 cm, then the distance between t and s is .5 cm .

tc. Since rt(D) 5 D, D is on line __________ .

d. Name two segments with length equal to DC'.

DC D"C"__________ __________


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Skills Objective DIn 1 and 2, draw the indicated image.

1. rp 8 rm(THE) 2. rs 8 r, (TRUE)

Properties Objectives F and GIn 3 and 4, multiple choice.

3. Rotations preserve d(a) collinearity and betweenness. (b) distance and angle measure.

(c) orientation. (d) all of the above.

4. If F is the image of G after successive reflections bacross lines which meet at a 478 angle, then

(a) G has been rotated 478 or -478. (b) G has been rotated 948 or -948.

(c) G has been translated 47 units. (d) G has been translated 94 units.

5. In the figure at the right, m∠ COL 5 55. Considerthe rotation which maps DABC onto DA" B"C".

a. The direction of the rotation is

.clockwise

b. Give the magnitude of the rotation. -1108

c. Name the rotation as a composite of two reflections.rt 8 rs

d. A rotation of what magnitude e. Name the rotation in Part d as amaps DA" B"C" onto DABC? composite of reflections.

-2508 (or 1108) rs 8 rt (or rt 8 rs )_________________________ _________________________

6. In each figure at the right s ⊥ p, s⊥ q. Nameeach composite of reflections as a rotation R,a translation T, both B, or neither N.

T Ra. rq 8 rp ________ b. rp 8 rs ________

B Rc. rp 8 rp ________ d. rq 8 rs ________


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Skills Objective CIn 1–3, draw and label the translation image ofthe figure determined by the indicated vector.

1. 2. 3. →CB

4. Draw and name a vector for thetranslation that maps ABCDonto A' B'C'D'.

Sample: →DD'

Representations Objective KIn 5 and 6, use the vector described by the orderedpair (-7, -10).

-7 -105. Name its a. horizontal component. _______ b. vertical component. _______

6. Find the image of the given point (-10, 0) (p 2 7,under translation by this vector. a. (-3, 10) _________ b. ( p, q) _________

q 2 10)7. The image of point A under translation by the vector

(-17, 33) is (-30, 29). What are the coordinates of A? (-13, -4)

8. Draw the image of PENTA under

the translation with vector (3, -1).


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Skills Objective CIn 1 and 2, draw the indicated glide reflection.

1. Draw G(ABCDE) 5 T 8 rp where

T is determined by →v.

2. Draw G(PQRS) 5 rm 8 T where

T is determined by→RQ.

3. DABC is the image of DRSTunder a reflection over line ,followed by a translation.Draw a translation vector.


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39

4. On the grid at the right, drawH(PQRS) 5 T 8 ry-axis where T has

the vector (-3, 2).


5. Name two isometries that do not preserve orientation.

reflection glide reflection___________________________ ___________________________

6. If you know that an isometry is the Sample:composite of three reflections, whatisometry might this composite be?Draw one possible situation foryour answer.

glide reflection

In 7 and 8, draw reflecting lines of an isometry Samples are that is the composite of three reflections given.7. and is a glide reflection. 8. and is not a glide reflection.

Uses Objective HIn 9–12, name the type of isometry that maps one letter onto the other.

9. 10. 11. 12.

reflection glide translation rotation____________ ____________ ____________ ____________

reflection

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Vocabulary

1. What is another name for an isometry? congruencetransformation

2. A treble clef is pictured at the right.

a. Draw a figure which is b. Draw a figure which isdirectly congruent to oppositely congruent tothe treble clef. the treble clef.

Uses Objective JIn 3–6, tell whether the performed task uses theidea of congruence. Explain why or why not.

3. blowing up a balloon Sample explanations are given.No; the size of the balloon is notpreserved.

4. breaking a vase

No; the vase is separated into parts.

5. flipping a coin

Yes; the size and the shape of the coinare preserved.

6. cutting cookies with a cookie cutter

Yes; the size and the shape of thecookies are preserved.


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VocabularyIn 1–4, true or false. Inthe diagram at the right,T(ABC) 5 XYZ.

1. The side corresponding to AC is XZ. true

2. The side corresponding to XY is BC. false

3. The angle corresponding to ∠ A is ∠ Z. false

4. The angle corresponding to ∠ XYZ is ∠ ABC. true

Skills Objective A

5. Suppose r, (POST) 5 PMAN, PO 5 7 cm,OS 5 17 cm, ST 5 5 cm, and TP 5 16 cm.Which side length in PMAN is 5 cm?

AN______________

6. rm 8 rn (WHY ) 5 NOT, where m and n intersect,at a 608 angle. m∠ W 5 67, m∠ H 5 30, andm∠ Y 5 83.

a. Which angle in nNOThas measure 67? ∠ N

b. If NT 5 16 cm, whichside in nWHY has thesame length? WY


7. nBUG > nFLY. List six pairs of congruent parts.

∠ B > ∠ F ∠ U > ∠ L ∠ G > ∠ Y_____________ _____________ _____________

BU > FL UG> LY GB> YF_____________ _____________ _____________

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Skills Objective A

1. In the figure at the right, supposeMD > DL, O is the midpoint of MD,and E is the midpoint of DL.If MD 5 28j, find each length.

28j 14j 14j 42ja. DL _______ b. OD _______ c. DE _______ d. ME _______

2. In the figures at the right, suppose

∠ ABL > ∠ MIS,→BE bisects

∠ ABL, and→IT bisects ∠ MIS.

If m∠ ABE 5 2.5t, find eachmeasure.

2.5t 5ta. m∠ MIT _______ b. m∠ MIS _______

Properties Objective EIn 3 and 4, name the property of congruence illustrated.

3. ∠ TOP > ∠ TOP Reflexive

4. If PO > TL and TL > MN, then PO > MN. Transitive

5. You are given only that ∠ COP > ∠ TOL. Samples:Draw a possible diagram. What can youconclude about ∠ COP and ∠ TOL?

∠ COP and ∠ TOL arecongruent and have thesame vertex.

6. Do the statements PQ > RS and PQ 5 RS mean thesame thing? Why or why not?

Yes; by the Segment Congruence Theorem,two segments are congruent if and only if they have the same length.

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Properties Objective EIn 1–3, multiple choice. Choose the justificationwhich allows the given conclusion.

1. If PQ 5 PT, then PQ > PT. d(a) CPCF Theorem (b) definition of midpoint

(c) definition of congruence (d) Segment Congruence Theorem

2. If →RZ is the bisector of ∠ XRT, then ∠ XRZ > ∠ ZRT. a

(a) definition of angle bisector (b) Corresponding Angles Postulate

(c) CPCF Theorem (d) Vertical Angles Theorem

3. If nABC > nXYZ, then ∠ A > ∠ X. a(a) CPCF Theorem (b) Corresponding Angles Postulate

(c) definition of congruence (d) Angle Congruence Theorem

4. In the diagram at the right, points A, B, and C areon (O. OB is the perpendicular bisector of AC.Provide the justification for each conclusion.

a. AP > CP def. perp. bisector

b. OA > OC def. circle

c. OP > OP Reflex. Prop. Congr.

d. ∠ APO > ∠ BPCVert. Angles Thm.

5. Given: r, 8 rm (ABCD) 5 EFGH, where , // m.To prove: ABCD > EFGH.

Sample: ABCD> EFGH by the definition of congruence.

6. Given: O is the midpoint of MP. P is the midpoint of OE.

a. To prove: MO > OP.

MO > OP by the definition of midpoint.b. To prove: OP > PE.

OP > PE by the definition of midpoint.c. Use the conclusions of Parts a and b to prove MO > PE.

MO > PE by the Transitive Prop. of Congr.

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Skills Objective B

1. Construct an equilateral triangle 2. Construct the circle which passeswith side QL. Sample: through the points given.Sample:

Skills Objective CIn 3–5, use the figure at the right.

3. Name two pairs of alternate interior angles.

∠ 2, ∠ 3 ∠ 6, ∠ 7____________________ ____________________

4. If m∠ 6 5 128, find as many other angle measuresas possible.

m∠ 15m∠ 25m∠ 35m∠ 45m∠ 52;

m∠ 55m∠ 75m∠ 85128

5. If m∠ 2 5 39 1 t, find the measure of each angle.

391t 1412t 391ta. ∠ 3 ____________ b. ∠ 5 ____________ c. ∠ 4 ____________


6. Suppose m∠ 2 5 15x 1 4 and m∠ 3 5 11x 1 15.

, or 2.75 45 , or 45.25x 5 ____________ m∠ 2 5 ____________

7. Suppose m∠ 5 5 20y 1 1 and m∠ 3 5 12y 1 3.

5 , or 5.5 69y 5 ____________ m∠ 4 5 ____________

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Properties Objective FIn 8–10, complete the proof by givingthe argument.

8. Given: MO > TO and TE > TO.To prove: MO > TE.

MO > TO and TE > TO is given.By the Transitive Property ofCongruence, MO > TE.

9. Given: m // n and ∠ 1 > ∠ P.To prove: ∠ 2 > ∠ P.

m // n and ∠ 1 > ∠ P is given.∠ 2 > ∠ 1 by the // lines ⇒AIA > Thm. Thus ∠ 2 > ∠ Pby the Transitive Prop. of Congruence.10. Given:

→SR bisects ∠ PSO and→SO bisects ∠ RSF.

To prove: ∠ PSR > ∠ FSO.

Since→SR bisects ∠ PSO and

→SO

bisects ∠ RSF (Given), ∠ PSR > ∠ RSO and∠ RSO > ∠ FSO (definition of angle bisector).Therefore, ∠ PSR > ∠ FSO (Transitive Prop. ofCongruence).

Uses Objective I

11. At the right is a diagram of a lightray traveling through a glass opticalfiber. If the ray makes a 308 anglewith the bottom of the fiber, find x, the measure of the angle at the top of the fiber. Justify your answer.

x 5 30; the two labeledangles are alternate interior angles andare congruent by the // lines ⇒ AIA> Thm.

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Skills Objective C

1. m is the ⊥ bisector of AB.

a. Name 2 pairs of congruent segments.

AC > BC PA > PB_____________ _____________

BPb. If AP 5 10, then _____________ 5 10.

5c. If AB 5 10, then AC 5 _____________ .

2.→RQ is the ⊥ bisector of MP. RM 5 3a,MQ 5 2b, RP 5 14, and QP 5 12.

, or 4.6 6a 5 ____________ b 5 ____________

3. Suppose ←→AC is the ⊥ bisector of XY. B is on

←→AC. AX 5 12t 1 1, AY 5 8t 1 15, and BX 5 8t.Find BY.

BY 5 28

Properties Objective GIn 4 and 5, supply the justification for eachstep of the proof.

4. Given: rt(A) 5 C and rt(B) 5 D.

To prove: AB > CD.

0. rt(A) 5 C;rt(B) 5 D Given

1. rt(AB) 5 CD Figure Refl. Thm.

2. AB > CD def. of congruence

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5. Given:←→TR is the ⊥ bisector of IA.

To prove: nTRA > nTRI.

0.→TR is the ⊥ bisector of IA Given

1. r←→TR

(I ) 5 A def. of reflection

2. r←→TR

(R) 5 R; r←→TR

(T ) 5 T def. of reflection

3. r←→TR

(nTRA) 5 nTRI Figure Refl. Thm.

4. nTRA > nTRI def. of congruence

In 6 and 7, complete the proof by writingan argument.

6. Given: r,(A) 5 D and r,(B) 5 E.To prove: nACB > nDCE.

r<(A) 5 D and r<(B) 5 E (Given), sor<(C) 5 C (def. of refl.). Thenr<(nACB) 5 nDCE (Fig. Refl. Thm.),and nACB > nDCE (def. of congr.).7. Given: JH is the ⊥ bisector of GI.

To prove: ∠ G > ∠ I.

Since JH is the ⊥ bisector of GI(Given), r←→

JH(G) 5 I, r←→

JH(H ) 5 H, and

r←→JHJ 5 J (def. of refl.) Then r←→

JH(nGHJ) 5

nIHJ (Fig. Refl. Thm.), and nGHJ > nIHJ (def. of congr.). Finally, ∠ G > ∠ I (CPCF Thm.)Uses Objective I

8. Use the street map at the right. The shortestdistance from Alexa’s house (A) to the postoffice (P) is the same as the shortest distancefrom the post office to the library (L). A, P, and L are on the same street. Explain why theshortest distance from Alexa’s house to theschool (S) is the same as the shortest distancefrom the school to the library.

Assuming SP ⊥ AL, by the ⊥ Bisector Thm. SA 5 SL.

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1. In nABC, there is 2. Segment AC is the only one bisector unique segmentof ∠ BAC. connecting A and C.

true true______________ ______________

3. There is a unique 4. It can be assumed that bisector of PE in in QUAD diagonalPENTA. QA bisects ∠ DQU.

false false______________ ______________

5. In nSCA, a student wished to draw as an auxiliary

figure the ray→SL which bisects ∠ CSA and is ⊥ to

CA. Is this possible? Why or why not? If youwish, draw a picture to support your explanation.

Sample: This is possible only if→SL is the ⊥ bisector of CA.

6. In the figure at the right, how many diametersof (P are parallel to ,? Justify your answer.

One; by Playfair’s ParallelPostulate, through a pointnot on a line, there isexactly one line parallelto the given line.

Culture Objective J

7. Who was first known to suggest the uniquenessof a line parallel to a given line through a pointnot on the given line, known also as Playfair’s Parallel Postulate? Proclus

8. Non-Euclidean geometries are applied in whatwell-known physical theory? Einstein’s Theory

of Relativity


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Skills Objective DIn 1–4, find the sum of the measures of theinterior angles of the figure.

180 9001. scalene triangle ____________ 2. heptagon ____________

1440 25203. decagon ____________ 4. 16-gon ____________

5. For the figure at the right, find each measure.

18 126a. a ____________ b. m∠ QPR ____________

6. Find the measure of each angle of the pentagon.

90 148.4a. ∠ A ____________ b. ∠ B ____________

90.8 90.8c. ∠ C ____________ d. ∠ E ____________

7. The measures of the angles of a triangle are in the extendedratio 8:6:2. Find the measure of the largest angle. 90

8. In triangle ABC, m∠ A 5 80. The measure of ∠ B is 15 more than 9 times the measure of ∠ C. Find m∠ B591.5 m∠ C58.5the measure of ∠ B and of ∠ C. ________________ ________________

Culture Objective J

9. Why did Gauss measure the angles of the triangle whosevertices were three mountaintops?

Gauss was trying to verify that theTriangle-Sum Theorem and other theoremsin Euclidean geometry were valid forlong distances.


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Skills Objective AIn 1–3, draw the symmetry line(s) for the figure.

1. 2. 3.


4. For every figure P, if r,(P) 5 P', then r,(P' ) 5 P . true

5. One symmetry line for PR is the ⊥ bisector of PR. true

6. Any line intersecting a circle at two different falsepoints is a symmetry line for that circle.

7. If r,(ABCD) 5 PQRS, then AC > PR. true

8. m and n are symmetry lines for polygonEFGHIJKL at the right.

a. Name three anglescongruent to ∠ G.

b. rn(GHLEF ) 5 IHLKJc. Must LK 5 KJ? Explain your answer.

No; KG←→

is not given as a symmetry line.

9.→PT bisects ∠ EPC.

a. r →PT

( →PC ) 5

→PE

b. What theorem justifies your answer to Part a?

Angle Symmetry Theorem

Uses Objective IIn 10–12, draw the symmetry line(s) for the figure.

10. 11. 12.

∠ E ∠ K ∠ I

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VocabularyIn 1–3, use nNPO at the right.

1. Name its base. PN.2. Name the base angle(s). ∠ P, ∠ N

3. Name the vertex angle(s). ∠ O

Skills Objective AIn 4 and 5, use nNPO above.

4. Draw the triangle’s symmetry line. 5. Mark a pair of conguent angles.

Skills Objective BIn 6 and 7, draw an example of the figure. Markthe congruent sides and congruent angles.

6. an isosceles triangle 7. an acute isosceles triangle

Skills Objective CIn 8 and 9, find the indicated measures.

8. m ∠ Q 5 76 9. m ∠ O 5 132

m ∠ P 5 28 m ∠ T 5 24

m ∠ R 5 24

10. nISO is isosceles with base IO.

a. If m ∠ I 5 (40 2 3x) and m∠ O 5 (5x 2 8), find x and the measure of each angle.

x 5 060 ∠ I 5 .22. ∠ O 5 .22. ∠ S 5 136x 5 ∠ I 5 ∠ O 5 ∠ S 5

b. If IS 5 2z 2 2, SO 5 z 1 5, andOI 5 4z 2 6, find z and each length.

z 5 070 IS 5 .12. SO 5 .12. OI 5 .22.z 5 IS 5 SO 5 OI 5


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11. Draw a triangle with three symmetrylines. Mark the symmetry lines.

12. nMNO is isosceles with vertex M and MP ⊥ NO.Name all pairs of congruent angles and segments.

MN > MO, NP > OP, ∠ N > ∠ O,

∠ MPN > ∠ MPO, ∠ NMP > ∠ OMP

Properties Objective HIn 13 and 14, complete the proof by givingthe argument.

13. Given: nABL is isosceles with base AB;nBLE is isosceles with base LE .

To prove: AL > BE.

Since nABL is isosceles (given),LA > LB (definition). Likewise,because nBLE is isosceles, LB > EB. Then,LA > EB (Transitive Property of Congruence).14. Given: nMIC is isosceles with vertex angle M;

nHAC is isosceles with vertex H.

To prove: ∠ I > ∠ A.

Conclusions Justifications1. MI. > MC; def. isos. n

HC > HA2. ∠ I > ∠ MCI; Isos. n base ∠ s thm.

∠ HCA > ∠ A3. ∠ MCI > ∠ HCA Vertical ∠ s Thm.4. ∠ I > ∠ A Trans. Prop. >

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Vocabulary

1. Define each term. quad. with 2 distinct a. kite pairs of consecutive sides >

b. rhombus quad. with all 4 sides >

c. trapezoid quad. with at least 1 pair of sides //

d. parallelogram quad. with both pairs of oppositesides //

Skills Objective BIn 2–4, draw a polygon satisfying thegiven conditions.

2. an isosceles 3. a pentagon that 4. a kite that is notright triangle is equiangular but a parallelogram

not regular

Skills Objective DIn 5 and 6, use parallelogram ABCD at the right.

5. If m ∠ D 5 4x, give the measure of each angle.

a. ∠ A 180 2 4x b. ∠ B 4x6. If AB 5 22 and AD 5 15, find each length.

a. BC 15 b. CD 22

Properties Objective GIn 7–9, use the markings on the quadrilateral togive it as specific a name as possible.

7. 8. 9.

__________________ __________________ __________________square rectangle isos. trapezoid


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Samples are given.

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10. Complete the argument below.

Given: Quadrilateral ABCD with AD ⊥ DC and BC ⊥ DCTo prove: ABCD is a trapezoid.


0. Quadrilateral ABCD; GivenAD ⊥ DC; BC ⊥ DC

1. AD // BC Two Perpendiculars Thm.

2. ABCD is a trapezoid. definition of trapezoid


Given: PO // AR and PA > OT. DOTR isisosceles with vertex O.

To prove: PORA is an isosceles trapezoid.

Sample: PORA is a trapezoidbecause PO // AR (given and def. of trapezoid).Then PA > OT (given) and OT > OR (def. ofisosceles triangle). So, PA > OR (TransitiveProp. of Congruence), and PORA is anisosceles trapezoid (def. of isosceles trapezoid).Representations Objective K

12. a. Draw a hierarchy relating quadrilateral,kite, square, trapezoid, and rhombus.

b. True or false. All kites are rhombuses.

falsec. True or false. Every square is a kite.

true

13. Draw a hierarchy relating quadrilateral,square, parallelogram, trapezoid, and isosceles trapezoid.

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rhombustrapezoid kite

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Skills Objective AIn 1 and 2, use COAL, a kite with ends C and A.

1. Draw the symmetry line for COAL.

2. a. Draw OL . b. OL ⊥ CA

c. Name two congruent angles. ∠ O, ∠ L

Skills Objective D

3. Use kite KITE with ends I and E. If m∠ 1 5 29and m∠ KIT 5 88, find each angle measure.

4. CORK at the right is a rhombus. Find each lengthand each angle measure.


5. Use the markings to give as specific a name as possible to each quadrilateral.

a. rhombus

b. kite



Given: Isosceles nTRI with base TI andisosceles nTAI with base TI.

To prove: TRIA is a kite.

Sample: It is given that nTRI andnTAI are isosceles with base TI.. By thedefinition of isosceles triangle, TR > IR and TA > IA. Thus, TRIA is a kite by definition.

OR 16

m∠ 2 52

m∠ KCO 104

RK 16

m∠ 3 38

m∠ COR 76

m∠ KTR 90

m∠ 4 52

m∠ 2 29

m∠ KET 58

m∠ 3 90

m∠ 5 44

m∠ 6 46

m∠ 7 61m∠ 4 44

m∠ ITE 107


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Skills Objectives A and D

1. Suppose TRAP is a trapezoid with bases TR andAP, m ∠ P 5 45, and m ∠ R 5 93.Find m ∠ A and m ∠ T.

m ∠ A 87 m ∠ T 135

2. a. Draw the symmetry lines for rectangle RECT.

b. If m ∠ REC 5 28 2 2a, find a. a 5 -31

3. In ABCD, m ∠ D 5 3y 1 2, m ∠ C 5 142 2 4y,m ∠ A 5 4z 1 10, m ∠ B 5 145 2 z. Find m ∠ Band m ∠ D.

m ∠ B 118 m ∠ D 62

Properties Objective GIn 4 and 5, true or false.

4. Every rectangle is an isosceles trapezoid. true

5. A square has four symmetry lines. true



Given: MP 5 NO and m∠ 1 5 m∠ N.To prove: MNOP is an isosceles trapezoid.

Conclusions Justifications0. MP 5 NO; Given

m∠ 1 5 m∠ N1. MN // PO AIA > ⇒ // lines2. MNOP is an definition of

isosceles trapezoid isosceles trapezoid


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Skills Objective AIn 1–3, if the figure has n-fold rotation symmetry,find n and mark the center of symmetry.

1. 2. 3.

rectangleequilateral triangle

n 5 2 n 5 3 n 5 4_____ _____ _____4. If a figure has 10-fold rotation symmetry, then the

least positive magnitude of a rotation that will mapthe figure onto itself is ....?..... 368

5. Draw a quadrilateral with the given symmetry(ies).

a. reflection and rotation b. neither reflection nor rotation

Uses Objective EIn 6–8, true or false.

6. All kites possess reflection symmetry. true

7. A rectangle has reflection symmetry but not rotation symmetry. false

8. All trapezoids are reflection-symmetric. false

9. Name the quadrilaterals with both reflection and rotation symmetry.

rectangles, squares, rhombuses

Uses Objective IIn 10 and 11, the object has n-fold rotationsymmetry. Find n.

10. 11.


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Samplesare given.

ancient Celtic design

n 5 2Ashanti box lid, Ghana

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Skills Objectives A and BIn 1–3 draw the figure and any symmetry lines.

1. a regular triangle 2. a pentagon that is 3. a quadrilateralequilateral but not equiangular but notregular regular

In 4 and 5, locate the center of symmetry.

4. 5.

Skills Objectives C and D

6. Find the measure of each angle of a triangle formedby three adjacent vertices of a regular 15-gon. 156, 12, 12

7. Find the measure of one interior angle of a regular

a. convex octagon. 135 b. convex 26-gon. ≈166.15


8. a. Name the center of rotationof the nonagon at the right. O

b. m ∠ ROQ 5 40

c. m ∠ RQO 5 70 d. m ∠ ROU 5 120



Given: ABCDEF is a regular hexagon.To prove: nBDF is equilateral.

Sample: By the def. of reg. polygon,AB > BC > CD > DE > EF > FA and ∠ C >∠ E > ∠ A. By the SAS Congr. Thm., nBCD >nDEF > nFAB. So, BD > DF > FB by the CPCFThm., and nBDF is equilateral by definition.


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Vocabulary

1. Multiple Choice. A chord of a circle is d(a) a line through the center of the circle.

(b) a segment which intersects the circle at exactly one point.

(c) a ray through two points on the circle.

(d) a segment that connects two points on a circle.

2. True or false. In a circle, the shorter the chord, truethe shorter the arc.

Uses Objective JIn 3 and 4, consider a round-robin match for11 teams in which games are played each week.

3. a. Mark the 11 teams on the circle at the right.

b. Draw the chord between two non-adjacentteams. Then draw all the chords parallel tothis chord. Write the first set of pairings.

1–10, 2–9, 3–8, 4–7, 5–6c. Rotate the chords of a revolution.

Write the second set of pairings. 11–9,1–8, 2–7, 3–6, 4–5

4. Continue until all the teams have played each other. Thenanswer the following questions.

a. How many weeks are needed? 11 weeks

b. How many individual games are needed? 55 games

c. How many byes are needed? 11 byes

d. How many weeks would be needed for twelve teams? 11 weeks

5. a. Write a schedule for a round-robin tournament for 6 teams.

1–6, 4–3, 2–5; 1–5, 2–4, 3–6; 1–4, 2–3, 5–6; 1–3, 5–4, 2–6; 1–2, 3–5, 4–6

b. How many weeks are needed? 5 weeks

c. How many individual games are needed? 15 games

d. How many byes are needed? no byes

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Skills Objective AIn 1–6, use an automatic drawer or other drawingtools. Draw the triangle with the given conditions.

SamplesThen tell whether you think all triangles with these are given.conditions are congruent.

1. nGHI with m∠ G 5 90, 2. nCAT with CA 5 2 cm,m∠ I 5 30, and m∠ H 5 60 AT 5 2 cm, and m∠ A 5 40

no yes____________ ____________

3. nTUP with m∠ T 5 50, 4. nLCH with LC 5 4 cm,m∠ U 5 100, and UP 5 1.50 CH 5 3.5 cm, and m∠ C 5 50

yes yes____________ ____________

5. nSMH with SM 5 4 cm, 6. nBAT with m∠ T 5 40,SH 5 4 cm, and MH 5 7 cm BA 5 3 cm, and AT 5 4.5 cm

yes no____________ ____________

7. Mrs. Liu challenged her class to construct a uniquetriangle with sides measuring 2 cm, 3 cm, and 6 cm.How would you respond to her challenge?

Sample: Since 2 1 3 < 6, there is no such triangle.

A T

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Skills Objective AIn 1–4, tell whether the conditions are sufficient toguarantee congruent triangles. Justify your answer.

1. ST 5 6 in., TU 5 4 in., m∠ T 5 110 yes; SAS > Theorem

2. m∠ S 5 25, m∠ T 5 45 no; no AAA > Theorem

3. m∠ U 5 70, m∠ T 5 46, ST 5 6.4 cm yes; AAS > Theorem

4. ST 5 20, SU 5 40, m∠ U 5 40 no; no SSA > Theorem

Properties Objective CIn 5–8, tell whether the triangles are congruent.If yes, write a congruence statement to indicatecorresponding vertices.

5. 6.

yes; nLAP > nORP yes; nMEN > nNAM

7. 8.

no yes; nRUT > nGUC

9. Two triangles are shown at theright. What additional informationis needed to show congruence?

AC > DF or ∠ B > ∠ E

Uses Objective I

10. Triangular supports for a coldframe were constructed asshown. Why are all the supportsthe same size and shape?

Sample: so that the frame is supported rigidly.

A

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6"

4'

Sample statements are given.

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Properties Objectives D and EIn 1 and 2, fill in the blanks to complete the proof.

1. Given: m∠ PLT 5 90, m∠ LTO 5 90,and LP > TO.

To prove: nLTO > nTLP.

Argument:


0. m∠ PLT 5 90, m∠ LTO 5 90, GivenLP > TO

1. ∠ PLT > ∠ LTO def. of congruence

2. LT > LT Reflexive Prop. of >

3. nLTO > nTLP SAS Congruence Thm.

2. Given: LU > UE and JU > UI.To prove: JE > LI.

Argument:


0. LU > UE; JU > UI Given

1. ∠ JUE > ∠ IUL Vertical Angles Thm.

2. nJUE > nIUL SAS Congruence Thm.

3. JE > LI CPCF Theorem

In 3–6, write a proof argument. Samples are given.3. Given: W is the midpoint of OT and ∠ O > ∠ T.

To prove: ∠ B > ∠ I.

Argument:

Since W is the midpoint of OT,OW > TW by definition. ∠ BWO >∠ IWT by the Vertical Angles Theorem. By the ASA Congruence Theorem, nBWO > nIWT. Finally, ∠ B > ∠ I by the CPCF Theorem.


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4. Given: ∠ E > ∠ R and ∠ EZB > ∠ RBZ.To prove: ZE > BR.

Argument:

Conclusions Justifications0. ∠ E> ∠ R; Given

∠ EZB> ∠ RBZ1. BZ > ZB Reflexive Property of

Congruence2. nBZE > nZBR AAS Congruence Thm.3. ZE > BR CPCF Thm.

5. Given: (O and AB > CD.To prove: ∠ COD > ∠ AOB.

Argument:

By the definition of circle,all radii are congruent, so OD > OBand CO > AO. By the Reflexive Propertyof Congruence, CD > AB. By the SSSCongruence Theorem, nCOD > nAOB.Thus, ∠ COD > ∠ AOB by the CPCF Theorem.

6. Given: Isosceles trapezoid TRAP with bases PTand AR; M is the midpoint of AR.

To prove: PM > TM.

Argument:

Conclusions Justifications0. Isosceles trapezoid Given0. TRAP with bases

AR and PT; M is the midpoint of AR.

1. AM > RM Def. of midpoint2. ∠ A > ∠ R Def. of isos. trapezoid3. AP > RT Isos. Trapezoid Thm.4. nAPM > nRTM SAS Congruence Thm.5. PM > TM CPCF Theorem

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VocabularyIn 1 and 2, redraw the figures separating thepairs of overlapping triangles. Label yourdrawings and mark the shared parts.

1. 2.

Properties Objectives D and EIn 3–5, provide the argument for the proof. Samples are given.3. Given: ∠ P > ∠ T and PS > TQ.

To prove: nPSR > nTQR.

Conclusions Justifications0. ∠ P > ∠ T; PS > TQ Given1. ∠ R > ∠ R Reflexive Property

of Congruence2. nPSR > nTQR AAS Congruence Thm.4. Given: JULIO is a regular pentagon.

To prove: JL > IU.

By definition, JU > IL and∠ JUL > ∠ ILU. By the ReflexiveProp. of Congruence, UL > LU. Then,nJUL > nILU by the SAS CongruenceThm. and JL > IU by the CPCF Theorem.5. Given: PS > QR and ∠ PSR > ∠ QRS.

To prove: ∠ 1 > ∠ 2.

Conclusions Justifications1. SR > RS Reflexive Prop.

of Congruence2. nPSR > nQRS SAS Congruence Thm.3. ∠ 1 > ∠ 2 CPCF Theorem

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Skills Objective AIn 1–4, determine whether all triangles withthe same measures as the given triangle arecongruent. Justify your answer.

1. no; two measures are

not sufficient.

2. a right triangle with hypotenuse yes; HL Congruence12 m and one leg 5 m Theorem

3. yes; SsA Congruence

Theorem

4. no; SsA CongruenceTheorem does not apply.

Properties Objective CIn 5–8, if the given triangles are congruent, justifywith a triangle congruence theorem. Otherwise, write not enough to tell.

5. 6.

not enough to tell not enough to tell___________________________ ___________________________

7. 8.

SsA Congruence Thm. HL Congruence Thm.___________________________ ___________________________


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Sample justifications are given

"7

5

2110°

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5"2"

15°

A CB

D

A CB

DAB > BD

A CB

DBD > AB

A CB

D

Properties Objectives D and EIn 9 and 10, write an argument to complete Samples are the proof.

given.9. Given: HR ⊥ HA, HR ⊥ DR, and HD > AR.

To prove: nHRA > nRHD.

Conclusions Justifications1. HR > RH Reflexive Property of

Congruence2. nHRA and nRHD Definition of right 2. are right ns. triangle 3. nHRA > nRHD HL Congruence Theorem10. Given: S is the midpoint of AY, ES > RS,

AY ⊥ EA, and AY ⊥ RY.

To prove: EA > RY.

Conclusions Justifications1. AS > YS Definition of midpoint2. nEAS and nRYS Definition of right 2. are right ns. triangle 3. nEAS > nRYS HL Congruence Theorem4. EA > RY CPCF Theorem

Uses Objective I

11. The volleyball net pictured at theright is supported by four ropes ofequal length, extending from thetops of the poles to the ground.The poles are the same length andare perpendicular to the ground.Explain why each rope makes the same angle x with the ground.

Sample: By the HL Congruence theorem, nACE > nACF >nBDG > nBDH. Then by the CPCF Theorem, ∠ AEC > ∠ AFC > ∠ BGD > ∠ BHD.

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Uses Objective J

1. Use the rectangle with length twice itswidth to create a brick-patio tessellationdifferent from the one shown at the right.

Sample:

2. A countertop is to be tiled with equilateraltriangles. Use the triangle at the right todemonstrate such a region.

Sample:

3. Use the design at the right. Outlinethe fundamental region used tomake the tessellation.

4. Explain why a regular hexagon can tessellate a regionand a regular heptagon cannot.

Sample: The measure of each angle of a regular hexagon is 120, and 360 is divisible by 120. The measure of each angle of a regular

heptagon is , and 360 is not divisible by .9007

9007


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Properties Objective FIn 1–3, true or false.

1. The center of rotation symmetry for a parallelogram trueis at the intersection of its diagonals.

2. All parallelograms have a symmetry line. false

3. In any parallelogram, two consecutive angles trueare supplementary.

In 4 and 5, use parallelogram PARL .

4. Locate the center of rotation symmetryof PARL.

5. Find x and y.

-1 6x 5 ____________ y 5 ____________

6. Use parallelogram ABCD at the right.

a. If MB 5 19 and AB 5 10, find as manyother lengths as you can.

MD 5 19, CD 5 10, BD 5 38b. If m∠ DAC 5 52 and m∠ BDC 5 22, find

as many other angle measures as you can.

m∠ ACB 5 52, m∠ ABD 5 22

Uses Objective K

7. Todd County in South Dakota has the approximate shape of a rectangle. Its northern border is about 40 miles long.

a. What is the length of its southern border? ≈ 40 milesb. What property of parallelograms

did you use in Part a? opposite sides >

8. Woodruff Park in Atlanta, Georgia isbounded by the four streets pictured at theright. Auburn is parallel to Edgewood and Peachtree is parallel to Prior. Peachtreemeets Edgewood at an angle of 648.

a. At what angles does Auburn meet Prior? 648b. What property of parallelograms is

illustrated by your answer to Part a? opposite angles >


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Properties Objective GIn 1–4, use the diagram. Are the markings asufficient condition for the quadrilateral to bea parallelogram? If your answer is yes,provide the sufficient condition.

1. 2.

no yes; 2 pr. opp. sides >____________________________ ____________________________

3. 4.

no yes; 1 pr. opp. sides // ____________________________ ____________________________and >

5. Consider this conditional: If three angles of a quadrilateralhave equal measures, then the quadrilateral is a parallelogram.

a. Draw an instance of b. Draw a counterexamplethe conditional. to the conditional.

Uses Objective K

6. Raul carefully measured the sides of his garden. Twosides measured about 309 and two others measured about199. Is the meadow a parallelogram? Why or why not?

No; the meadow could be a kite.

7. An expandable gate is constructed so that for eachquadrilateral region, AB 5 CD and AD 5 BC.

a. Explain why ABCD remains a parallelogram forany expansion of the gate.

Both pairs of sides remain >.

b. Explain why m∠ B always equals m∠ D.

Opposite angles of a parallelogram are >.


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Skills Objective B

1. a. Draw an exterior angle at each vertex Sample:of nABC at the right.

b. The sum of the angles 360that you drew in Part a is _________ .

2. 3.

78 59Find m∠ 1. _________ Find m∠ 2. _________

4. 5. Find x and m∠ ACB.

107Find m∠ 3. _________

73 22 29Find m∠ 4. _________ x _________ m∠ ACB _________


6. Use the diagram at the right.

a. What is the relationship m∠ 1 > m∠ Pbetween m∠ 1 and m∠ P? _________________

b. What theorem justifies that relationship?

Exterior Angle Inequalityc. If m∠ 1 5 5a 2 60 and m∠ N 5 2a, write an 5a 2 60 > 2ainequality to express the possible values of a. _________________

7. Use nRAG at the right.∠ Aa. Name the largest angle. ____________

∠ Rb. Name the smallest angle. ____________

8. Name the sides of nABC at the right in orderof length from shortest to longest.

BC AC AB____________ ____________ ____________


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Skills Objective AIn 1–5, give the perimeter of each figure.

1. a rectangle with length 5 cm and width 3 cm 16 cm

2. an equilateral triangle with one side of length 9 in. 27 in.

3. a square-shaped region of land mile wide 2 miles

4. a regular heptagon with one side of length 10 mm 70 mm

5. a regular pentagon with side (x 1 3) (5x 1 15) units

6. Pictured at the right is kite DEFG.If its perimeter is 48, what are thelengths of its sides?

6, 6, 18, 18 units

Uses Objective H

7. A billboard is 1.5 times as wide as it is high. If itsperimeter is 40 meters, how wide is the billboard? 12 meters

8. A tennis court is 36 feet wide and78 feet long. A fence is to be builtaround the court 10 feet from theedges of the court. How muchfencing will be needed?

308 feet

9. The fences surrounding an isosceles-triangle-shapedranch have a total length of 18 km. If the triangle’sbase is 4 km, find the lengths of the other sides. 7 km, 7 km

10. Flowers are to be placed at 1-foot intervals arounda rectangular garden. If the garden is 4 yards wideand 16 yards long, how many flowers will be needed? 120 flowers

12


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Skills Objective C

1. Rectangle RECT has dimensions 9 feet and 15 feet.

a. Find Area(RECT ) in square feet. 135 sq ft

b. What are the dimensions of RECT in yards? 3 yd, 5 yd

c. Find Area(RECT) in square yards. 15 sq yd

2. Find the area of a square with side length inch., or .5625, in2

3. A square has a perimeter of 20x.

a. Give an expression for the area of the square. 25x2

b. Evaluate the area for x 5 3. 225 units2

4. The area of a rectangle is 455 square centimeters andits length is 26 centimeters. Find its width. 17.5 cm


5. Draw three different rectangles with perimeter of30 units. Find the area of each.

Uses Objective I

6. How much carpet is needed to cover the floorof a room that is 12 feet by 16 feet? 192 sq ft

7. How many 8-yd-by-25-yd tarpaulins will be needed tocover a soccer field which measures 50 yd by 120 yd? 30 tarps


8. Find the perimeter and area of JKLMNO.

perimeter 26 units

area 34 units2

9163

4


A

(-2, 4)

L (2, 0)

M(2, -2)

N(5, -2)

K(-2, 0)

x

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(5, 4)O

7.5 un.

7.5 un.

56.25 un.25 un.

10 un.

50 un.2 3 un.12 un.36 un.2

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Skills Objective BIn 1–3, use the method in this lesson to estimatethe area of the irregular region. The side lengthof one small square is given.

1. 1 mile 2. .2 km 3. 50 yd

≈ 25 mi2 ≈ 1.5 km2 ≈ 86,250 yd2______________ ______________ ______________

4. Estimate the area of the kite.

a. Use the left-hand grid.

≈ 22 cm2______________

b. Use the right-hand grid.

≈ 17 cm2______________ side of squares 5 2 cm side of squares 5 1 cm

c. Which answer is more accurate? Explain.

b; the finer the grid, the better the estimate.

5. An irregularly shaped region appears on a computerscreen. 150,000 pixels are contained within the regionand 7,200 pixels are on the region’s boundary.

a. What is the area of the region (in pixels)? 153,600 pixels

b. If the entire screen is composed of 640 rows and480 columns of pixels, what percent of the screenis covered by the region? 50%

c. Find the area of the region with 150,000 pixels onthe region’s boundary and 7,200 pixels containedwithin the region. 82,200 pixels

d. What percent of a screen composed of 640 rowsand 480 columns of pixels is covered by this region? ≈ 27%


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Samples aregiven for 1–4.

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Skills Objective CIn 1–3, find the area of nABC.

1. 2. 3.

AD 5 9 cm, DB 5 5 cm,DC 5 2.5 cm Area (nACD) 5 60 m2

6 units2 28.75 cm2 12 m2______________ ______________ ______________

4. If Area (nDEF) 5 1134, find GE.

36 units______________


5. Explain how the formula for the area of a right triangleis related to the formula of a rectangle.

Sample: A rectangle is two right triangles, soits area is 2 ( bh) 5 bh.

Uses Objective I

6. On the boat shown at the right, the height of the mastis 18 feet and the length of the boom is 14 feet.

a. How much material is needed to construct a triangular sail for the boat? 126 ft2

__________

b. If the same amount of material is used for a sail for a boat with a 21-footmast, whatis the maximum length for its boom? 12 ft__________


7. A triangle has vertices (-3, 3), (2, 7),and (5, 3).

a. Draw the triangle on the grid atthe right.

b. Find the area of the triangle.

16 units2______________

12


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Skills Objective CIn 1–5, use the information in the drawingto find the area of the largest trapezoid.

1. 2. 3.

x2 units2 1176 units2 513 units2______________ ______________ ______________

4. 5.

6. The area of a rhombus is 232 cm2, and its perimeteris 64 cm. Find the length of its altitude. 14.5 cm

7. A trapezoid has an area of 120 in2. Its altitude Sample:measures 8 in. Give a possible pair of lengths forthe bases of the trapezoid. 14 in., 16 in.


8. Of parallelograms ABCD, EFCD, andGHCD, which has the greatest area? Howdo you know?

Sample: All have thesame area, as they

have the same base CDand the same height.


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Uses Objective I

9. The Canadian province ofSaskatchewan is shaped roughlylike an isosceles trapezoid whosedimensions, according toSaskatchewan’s Office of CentralSurvey and Mapping, are given onthe map at the right. Estimate thearea of Saskatchewan.

≈ 258,955 km2_____________________

10. A lawn-maintenance person canmow about 9,000 square feet ofgrass per hour. At this rate, howlong would it take for this person tomow the plot of land at the right?

3.5 hours_____________________

Representations Objective KIn 11 and 12, find the area of the region.

11. 12.

35 units2 1350 units2____________________ _____________________

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773 km

277 km

126 ft149 ft210 ft

290 ft

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x

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10

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20

30

40

10 20 30 40 50

y

x-1 2 4 6

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3

5

7(3, 6)

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(-1, 6)

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Skills Objective DIn 1–3, find the length of the missing side andthe area of the triangle.

1. 2. 3.

Skills Objective EIn 4 and 5, could the numbers be the lengthsof sides of a right triangle?

yes no4. 60, 80, 100 __________ 5. __________

6. Find the perimeter and the area of a 7. Find the perimeter and the area of right triangle with a hypotenuse of a rhombus with diagonals10 mm and one side 6 mm long. measuring 16 in. and 30 in.

24mm 68 in.perimeter __________ perimeter __________

68 in. 240 in2area __________ area __________

Uses Objective H

8. The infield of a baseball diamond is a square-shaped region 90 feet on a side. When a center-fielder throws a ball to home plate from 100 feetbehind second base, how long is the throw?

≈ 227 ft______________

9. Ramona found a letter on the corner of a rectangular park and decided to put it in amailbox which was on the opposite corner of the park. The perimeter of the park is 344 m, and it is 10 m longer than it is wide. How much farther is it for her to walk along the sidewalk than to walk directly to the mailbox?

≈ 50 m______________

Culture Objective L10. Identify the cultures that knew of the Pythagorean

Theorem more than 2000 years ago.

Greek, Babylonian, Indian

Ï3, Ï4, Ï5


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Skills Objective F

1. Multiple choice. Which expression shows the exact dcircumference of a circle with a diameter of 15 km?

(a) 7.5πkm (b) 47.1 km (c) 30πkm (d) 15πkm

2. Give the circumference of a circle with a diameterof 12 feet

12πft 38 fta. exactly. ____________ b. to the nearest foot. ____________

In 3–5, find the radius of a circle whose circumference is given.

59 ≈ 1.39 ≈ 2.23x3. 118π ___________ 4. 8.75 ft ___________ 5. 14x mm ___________

In 6–8, find the length of an arc with the givenmeasure on a circle whose radius is 18.

≈ 56.55° ≈ 37.70° ≈ 22.62°6. 180˚ ___________ 7. 120˚ ___________ 8. 72˚ ___________

9. In the circle at the right, OB 5 3.5 andCHB 5 2.75. Find mCHB.

≈ 45°___________

Uses Objective J

10. The earth’s orbit can be approximated by a circlewith a radius of about 93,000,000 miles. What isthe circumference of this circle? ≈ 584,340,000 mi

11. The gold for a certain wedding band costs $12per millimeter. How much more expensive is thewedding band for someone whose finger has adiameter of 17 mm than for someone whose fingerhas a diameter of 14 mm? ≈ $113


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Skills Objective FIn 1–3, estimate to the nearest tenth the area of the circle described.

1. a radius of 6 inches 2. a circumference of 3. a diameter of cm18πunits

113.1 in2 254.5 units2 15.7 cm2_________________ _________________ _________________


4. Which has the greater area, a circle with a diameter of 4 feetor a square with a side of 4 feet? Justify your answer.

Square; the square’s area is 16 square feet,while the circle’s area is about 12.6 square feet.

Uses Objective J

5. Suppose a baseball field is shaped like aquarter-circle with a radius of 350 feet asshown at the right. What is the area of theoutfield (the shaded region)?

≈ 88,111 ft2_________________

6. A cellular phone can transmit a call anywhere within a 50-mile radius from the transmission point.What is the area of the transmission region? ≈ 7854 mi2

7. A CD with a diameter of 12 cm is packed in a square casewhich measures 13 cm along a side. How much space isaround the CD?

≈ 55.9 cm2_________________

2Ï5


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Vocabulary

1. State the converse of the Flat Plane Assumption.

If a line containing two given points lies ina plane, the two points lie in the plane.

2. Define intersecting planes.

two different planes that have a point in common

Properties Objective FIn 3–5, state whether the figure is contained in aunique plane. State the Point-Line-Plane PostulateAssumption which justifies your answer.

3. all of Euclidean space No; given a plane in space, thereis at least one point in space not on the plane.

4. three collinear points No; the points determine a line,and the intersection of two planes is a line.

5. an isosceles triangle Yes; through three noncollinearpoints, there is exactly one plane.

In 6–9, match the situation with the Point-Line-PlanePostulate Assumption it most closely illustrates.

(a) Unique Line Assumption (b) Number Line Assumption

(c) Dimension Assumption (d) Flat Plane Assumption

(e) Unique Plane Assumption (f) Intersecting Planes Assumption

6. the stability of a tripod e

7. on a piece of notebook paper, there are cpoints other than those on the top edge

8. two walls that meet at a corner in a room f

9. on a flat green, a putt along only one path will aget a golf ball from its lie to the hole


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Skills Objective AIn 1–4, draw the figure.

1. a line which intersects a plane but 2. a segment perpendicular to both ais not perpendicular to the plane plane and a line parallel to the plane

3. a line which is perpendicular 4. a dihedral angle with measure 65to two planes


5. Use the figure at the right, in which , isperpendicular to plane R at O, and m andn are in plane R.

a. is , ⊥ m? Explain your answer.

Yes; m is in R and bydefinition , is ⊥ to any linein R that passes through ,.

b. is m ⊥ n? Explain your answer.

No; m and n intersect, but they are notnecessarily perpendicular.

6. Provide an argument for the proof.

Given: Points A, C, D and E lie in plane X.AC is a bisector of DE. PB is ⊥ toplane X.

To prove: DP 5 EP.

By def., DB 5 EB. By the Reflexive Prop. ofEquality, PB 5 PB. By def., ⊥ andm∠ PBD and m∠ PBE 5 90. So, nPBD > nPBEby SAS > Thm., and DP 5 EP by CPCF Thm.

DE PB


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1. a right cylinder 2. a right rectangular prism

3. a right triangular prism 4. an oblique pentagonal prism

Skills Objective D

5. Consider the triangular prism drawn at the right inwhich CA ⊥ AB.

5 unitsa. Find FE. ______________

b. What is the area of each lateral face?

40 units2, 32 units2, 24 units2

6. Use the oblique cylinder at the right.

9π ≈28.3units2a. Find the area of its base. ______________

≈ 10.6 unitsb. Find its height. ______________


7. a can of soup 8. a tent 9. a row of CD boxes thatleans in one direction

rt. cylinder rt. triangular oblique rect.______________ ______________ ______________

prism prism


A

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1. a right cone whose height is 2. a right hexagonal pyramidshorter than the radius of its base.

3. an oblique cone whose smaller 4. a truncated square pyramidangle to the base has measure 60

Skills Objective D

5. The right square pyramid at the right has lengthsas marked.

≈ 32.7 unitsa. Find its height. ______________

437.5 units2b. Find the area of one lateral face. ______________

6. In the regular pentagonal pyramid at the right,AC 5 9 and DC 5 8.

45 unitsa. Find the perimeter of its base. ______________

≈ 6.6 unitsb. Find its slant height. ______________


7. a flower pot truncated right cone

8. the Transamerica Building right square pyramid

9. a stalactite right cone


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Skills Objective A

1. Draw a hemisphere with a horizontalbase and a vertical plane section of it.

Sample:

Skills Objective BIn 2–5, sketch the plane section and name its shape.

2. parallel to the base 3. parallel to the bases ofof a square pyramid an oblique cylinder

square circle________________ ________________

4. perpendicular to 5. neither parallel tothe base of a right nor intersecting thecone and through bases of a regularthe vertex hexagonal prism

isosceles non-regular________________ ________________

triangle hexagon

Skills Objective D

6. The radius of a sphere is 14.4 in. What isthe area of a great circle of the sphere? ≈ 651.4 in2

7. A plane section is formed when a plane cutsperpendicularly through the diameter of thebase of the right cylinder pictured at the right.What is the area of the plane section?

150 units2

Uses Objective HIn 8 and 9, identify both the 3-dimensional figureand the kind of plane section described.

8. the middle layer of icing in a round birthday cake

right cylinder; circle

9. a florist’s oblique cut through a flower stem

right cylinder; ellipse


A

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5

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Properties Objective GIn 1–3, a figure is given. a. Tell if the figure has bilateral symmetry. b. If so, give the number of symmetry planes.

1. 2. 3.

right square pyramid coffee mug egg

yes yes yesa. ____________ a. ____________ a. ____________

4 1 infiniteb. ____________ b. ____________ b. ____________

number

4. A regular prism which has a hexagon for a base hashow many symmetry planes? 7

5. Draw a prism with no planes of symmetry.

Sample:

oblique prism withscalene triangle base

6. Draw a pyramid with one plane of symmetry.

Sample:

oblique triangular pyramid withisosceles triangle base


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Skills Objective CIn 1–3, a figure is given. Sketch views of the figurefrom a. the top. b. the front. c. the right side.

1. 2. 3.

a. ____________ a. ____________ a. ____________

b. ____________ b. ____________ b. ____________

c. ____________ c. ____________ c. ____________

Skills Objective EIn 6 and 7, name a surface with these views.

6. 7.

sphere truncated square________________________ ________________________

pyramid8. Use the given views of

the building.

a. How tall in stories isthe building?

2 storiesb. How long in sections is the c. Sketch the shape of

building from front to back? the building.

3 sections

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topsidefront

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VocabularyIn 1–3, give the number of faces, vertices andedges of the polyhedron. Describe the faces.

1. regular tetrahedron 4 faces which are congruentequilateral triangles, 4 vertices, 6 edges

2. regular dodecahedron 12 faces which are congruentregular pentagons, 20 vertices, 30 edges

3. regular icosahedron 20 faces which are congruentequilateral triangles, 12 vertices, 30 edges

Representations Objective JIn 4–6, sketch the figure that can be made fromthe given net.

4. 5. 6.

7. Draw a net for an oblique rectangular prism.

Sample:

8. Draw a net for a standard die pictured at the right, makingsure that the numbers of dots on opposite faces add to 7.

Sample:


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Uses Objective I

1. Draw a map with six regions that 2. Draw a map with six regions thatcan be colored with two colors. needs four colors to be colored.

Sample: Sample:


3. Explain one problem of using a Mercator-projection map.

Sample: the net for a Mercator-projection mapforms a cylinder rather than a sphere.

4. Which of the A-B-C-D properties does the map below preserve?

betweenness, collinearity


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Skills Objective AIn 1 and 2, give the surface area of the figure.

1. a right triangular prism with a 2. a right cylinder with circumferencebase of side lengths 3 cm, 4 cm, 2πinches and height 1 inches.and 5 cm, and a height of 4 cm

60 cm2 5π≈ 15.7 in2________________________ ________________________

3. Refer to the right cylinder at the right.


80π≈ 251.3 cm2


112π≈ 351.9 cm2

4. Refer to the box at the right.


156x2 units2


236x2 units2

Uses Objective H

5. a. A paint roller is 12 inches long andhas a radius of inch. What is itslateral area?

18π≈ 56.5 in2

b. Suppose the paint roller has a radiusof inch and is 18 inches long.What is its lateral area?

18π≈ 56.5 in2

6. A gift box measures 40 in. by 28 in. by 12 in. Can thebox be completely covered by a 30-ft2 roll of wrappingpaper? Why or why not?

Yes; the surface area of the box is ≈ 26.9 ft2

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Skills Objective BIn 1 and 2, give the surface area of the figure.

1. a right cone with a slant height 2. a right square pyramid with a slantof 12 and a radius of 6 height of 5 and a height of 4

108π≈ 339.3 units2 96 units2________________________ ________________________

3. Find the lateral and surface areas of the regularpyramid shown at the right.

6500 units2 9000 units2________________________ ________________________

4. The slant height of a regular square pyramidis 17 m and its lateral area is 544 m2.What is the side length of its base? 16 m

5. Find the lateral area of a right cone whoseheight is 15 m and whose radius is 8 m. 136π≈ 427.3 m2

Uses Objective H

6. How much paper would be needed to constructa birthday hat which is in the shape of a rightcone with radius of 2.5 inches and a slant heightof 5 inches? 12.5π≈ 39.3 in2

7. How much wood is needed to replace the entireroof of the gazebo at the right which is in theshape of a right pyramid on a regular octagon?

192 ft2________________________


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Skills Objective AIn 1–3, give the volume of the box with thegiven dimensions.

1. 4, 4, 4 2. 8 cm, 11 cm, 18 cm 3. 5.1 in., 8.8 in., 1 ft

64 units3 1584 cm3 538.56 in.3________________ ________________ ________________

4. Find the volume of a cube with side length p. p3 units3

5. Find the volume of a cube with side length ( p 1 1). (p313p213p11) un.3

In 6 and 7, give the volume and the surface areaof the box.

6. 7.

36 un.3 abc un.3V 5 ____________ V 5 ____________

72 un.2 2(ab 1S.A. 5____________ S.A. 5 ____________

ac 1 bc) un.2

Skills Objective CIn 8–10, give the cube root of the given numberto the nearest tenth.

7 1.5 12.68. 343 ____________ 9. 3.375 ____________ 10. 2,000 ____________

11. The volume of a cube is 60 cubic centimeters. Whatis the length of an edge, to the nearest tenth? 3.9 cm.

Uses Objective I

12. An air-conditioner manufacturer claims that a certainmodel can cool any room with volume no greater than2,000 cubic feet. Can it cool a room that measures14 feet by 12 feet by 9 feet? Explain your answer.

Yes; the volume of the room is 1512 ft.3

13. The volume of a box is 12,960 cm3. It is 24 cm longand 18 cm wide.

a. What is its height? b. What is its volume in cubic millimeters?

30 cm 12,960,000 mm3______________ ______________


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1. The dimensions of a box are x, 2x and 4x. Whenall dimensions are multiplied by 3,

a. what happens to its volume? It is multiplied by 27.

b. what happens to its surface area? It is multiplied by 9.

2. A box has dimensions 5 in., 8 in., and 10 in.

a. What are the volume and surface area of this box?

400 in3 340 in2V 5 ______________ S.A. 5 ______________

b. What are the volume and surface area if each dimension is multiplied by 4?

25,600 in3 5440 in2V 5 ______________ S.A. 5 ______________

3. The length and width of the bottom of a box are eachmultiplied by . How will its lateral area change?

It will be divided by 4.

Representations Objective JIn 4–6, expand.

4. r (2r 1 1)(3r 1 4) 5. (2y 1 2)(3y 1 9) 6. (x 1 2)3

6r 3111r 214r 6y2124y118 x316x2112x18_________________ _________________ _________________

7. Find the area of the rectangle. 8. Find the volume of the box.

(a 1 b)(a 1 c) 5 (x15)(y13)(z18) 5___________________________ ___________________________

a2 1 ab 1 ac 1 bc xyz18xy13xz124x15yz140y115z1120

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Skills Objective AIn 1–3, calculate the volume of the figure.

1. 2. 3.

18,849.6 un.3πr2h units3 760 units3 6000π≈___________ ______________ ______________

4. What is the radius of an oblique cylinder that has a volumeof 722πmm3 and a height of 8 mm? 9.5 mm

5. A right cylinder and a right prism with a square base eachhave a volume of 2205 cm3 and a height of 20 cm. Whichis greater, the diameter of the base of the cylinder or a sidelength of the base of the prism? Justify your answer.

diameter; diameter ≈ 11.8 cm, side length= 10.5 cm


6. Does Cavalieri’s Principle apply to the prismsat the right? Explain why or why not.

No; the bases are notcongruent so the slicesdo not have equal area.

Uses Objective I

7. Which holds more water, a 7-inch tall cylindricalglass with a diameter of 2.5 inches or a 4-inch tallcylindrical glass with a diameter of 3.5 inches? 4-in. tall glass

8. A horse trough is shown at the right.If it is filled to the top with grain,how many cubic feet of grain willthe trough hold?

6 ft3



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1. When deriving formulas for cones and cylinders fromthe basic formulas for conic and cylindric surfaces, what circle formulas can be used?

πr 2 2πr or πda. B 5 ______________ b. p 5 ______________

2. For each figure below, write a specific surface-area formula.

right cone right cylinder

πr 2 1 πr< 2πr2 1 2πrha. _________________ b. _________________

regularrectangular prism rectangular prism

2(<w 1 <h 1 wh) s2 1 2s<c. _________________ d. _________________

3. For some figures, the formula for lateral area is L.A. 5 ,p.

a. Draw two different figures b. Draw a figure for whichfor which this is true. this is not true.

Samples: Sample:

4. Give the formula for the surface area of a squarepyramid whose slant height is 3x and whose basehas a side length x 1 2. 7x2 1 16x1 4 un.2

5. Multiple choice. The general formula for the lateral area ofa right conic surface is L.A. 5 ,p. Which expression givesthe specific lateral area of a cone? d

(a) L.A. 5 πrh (b) L.A. 5 (2πr 2h)

(c) L.A. 5 πr, (d) L.A. 5 (2πr , )1122

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Name


Skills Objective BIn 1–5, find the volume of the figure.

1. square pyramid 2. right cone 3. rectangular pyramid

25s units3 132π≈ 414.7 units3 2304 units3______________ ______________ ______________

4. oblique pyramid with 5. an oblique cone whose base has atriangular base circumference of 4π

920 units3 18π≈ 56.5 units3______________ ______________

6. A regular square pyramid has a base with sides of length 7.5 m. If its volume is 225 m3, what is its altitude? 12 m

Uses Objective I

7. The sharpened end of a pencil has the shape of aright cone. The diameter of the base is 7 mmand the height is 22 mm. What is the volume ofthe whole pencil point?

≈ 89.83π≈ 282.22 mm3

8. A structure is composed of a cube-shaped baseof edge length 16 ft and a pyramid-shaped roof.In order for the roof to have the same volume asthe base, what must be the altitude of the roof?

48 ft



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Skills Objective DIn 1–3, draw the sphere with the given dimensionand find its volume.

1. radius 5 1 cm 2. diameter 5 2.5 cm 3. diameter 5 0.75 in.

π≈ 4.2 cm3 π≈ 8.2 cm3 π≈ .22 in3

In 4–6, give the radius of the sphere with the given volume.

4. 36π 5. 420π 6. 1234 ft3

3 units ≈ 6.8 units ≈ 6.7 ft


7. The diameter of Saturn is about 10 times that of Venus.How do their volumes compare?

Saturn’s volume is about 1000 times that of Venus.

Uses Objective I

8. A tank full of 15,000 in3 of air is to be used toinflate a beach ball with diameter 20 in., abasketball with diameter 9.5 in., and a volleyballwith diameter 8 in. Does the tank containenough air to fill all of the balls? yes

9. A globe with a 16-cm radius just fits into a cube-shaped box. The rest of the box is filled withpacking material. What is the volume of thepacking material? ≈ 15,611 cm3

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Skills Objective DIn 1–5, find the surface area for the sphere withthe given dimension.

1. radius 5 4 in. 2. diameter 5 0.28 km

64π≈ 201.1 in2 .0784π≈ .246 km2

3. circumference of a great circle 5 22πcm 484π≈ 1520.5 cm2

4. circumference of a great circle 5 16.44 m ≈ 86.0 m2

5. volume 5 288πin3 144π≈ 452.4 in2

6. A sphere has a surface area of 722 ft2. Find its radiusto the nearest hundredth of a foot. 7.58 ft


7. The diameter of Uranus is about 4 times that of Earth.How do the surface areas of the two planets compare?

Surface area of Uranus is ≈ 16 times that of Earth.

8. If a sphere’s radius shrinks to of its original size,what happens to its surface area?

It shrinks to of its original surface area.

Uses Objective H

9. A Hall-of-Fame baseball player wants to bronze thelast home-run ball he hit. The radius of a ball is1.5 in. Find the area of the surface to be bronzed. 9π≈ 28.3 in2

10. A concert shell, in the shape of a quarter of a spherewith a radius of 80 feet, is to be painted. If one gallonof paint covers 400 square feet, how much paint isneeded to cover the outside of the shell? 16π≈ 51gal.

11. The planet Mercury has a radius of about 2400 km.If a surveying satellite can take a picture whichcovers an area of 30,000 km2, what is the leastnumber of pictures that will need to be takento completely survey the surface? 2413 pictures

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Properties Objective DIn 1–3, use both given statements. a. What(if anything) can you conclude? b. What law(s)of reasoning have you used?

1. (1) If a triangle has two congruent sides, it is anisosceles triangle.

(2) In nABC, AB > BC.

a. nABC is an isosceles triangle.

b. Law of Detachment

2. (1) If a 5 4, then b 5 7.(2) If b 5 7, then c 5 9.

a. If a 5 4, then c 5 9.

b. Law of Transitivity

3. (1) If quadrilateral PQRS is a square, it is alsoa rectangle.

(2) Quadrilateral PQRS is a rectangle.

a. no conclusion

b.

Uses Objective HIn 4–6, use all the given statements. What(if anything) can you conclude?

4. (1) Maria practices the cello on every day whose name begins with the letter T.

(2) Today is Thursday.

Maria practiced the cello today.

5. (1) Sam eats tiramisu every year on his birthday.(2) Last Tuesday, Sam ate tiramisu.

no conclusion

6. (1) If the Central High Cougars win Friday’s game,they will win the championship.

(2) If Jaime pitches for Friday’s game, the Cougarswill certainly win the game.

(3) Jaime will pitch on Friday night.

The Cougars will win the championship.


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Properties Objective DIn 1 and 2, use both given statements. a. What (if anything)can you conclude? b. What law(s) of reasoning have you used?1. (1) If m∠ B is 90, then nABC is a right triangle.

(2) nABC is not a right triangle.

a. m∠ B ≠ 90 b. Law of the Contrapositive

2. (1) If 4 1 a 2 5 9, then a 5 6 . (2) a < 0

a. a 5 - b. Law of Detachment

Properties Objective EIn 3 and 4, a statement is given. a. Write itsconverse. b. Write its inverse. c. Write itscontrapositive. d. If the original statement istrue, which of b, c, and d are also true?3. In nLMN, if MN 5 5, then LM 1 LN > 5.

a. If LM 1 LN > 5, then MN 5 5.

b. If MN ≠ 5, then LM 1 LN ≤ 5.

c. If LM 1 LN ≤ 5, then MN ≠ 5.

d. c (contrapositive)

4. If it is cloudy, then it is raining.

a. If it is raining, then it is cloudy.

b. If it is not cloudy, then it is not raining.

c. If it is not raining, then it is not cloudy.

d. c (contrapositive)

Uses Objective HIn 5 and 6, use both given statements. What (if anything)can you conclude using the laws of reasoning?5. (1) Antonio lives in Stockholm.

(2) Stockholm is not located in Italy.

Antonio does not live in Italy.

6. (1) A dog has four paws.(2) My pet has four paws.

no conclusion

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1. If statement A or statement B is true, and statement Bis not true, then ....?.... must be true. statement A

2. If nRST is neither isosceles nor equilateral, what can you conclude?

nRST is a scalene triangle.


(1) WXYZ is either a rectangle or a trapezoid.(2) A quadrilateral is a rectangle if and only if it has four right angles.(3) WX is not perpendicular to XY.

a. WXYZ is a trapezoid.

b. Law of Ruling Out Possibilities

Uses Objective H


Kevin, Mel, and Ken each ate a different breakfast, consisting of eggs, cereal, or a bagel.

(1) Mel never eats eggs.(2) Kevin ate either cereal or a bagel.

a. Ken ate eggs for breakfast.

b. Law of Ruling Out Possibilities

5. Chris, Becky, Rob, and Jim are putting on a play andthey each have a different role from among these four: theScarecrow, the Lion, the Tin Man, and Dorothy. From theclues below, determine who plays which part.

(1) The scarecrow is not Becky, Rob, or Chris.(2) Chris is very good friends with Dorothy and the Lion.(3) Rob gave a costume to Dorothy.

Chris—Tin Man; Becky—Dorothy;Rob—Lion; Jim—Scarecrow


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Properties Objective DIn 1 and 2, statements p and q aregiven. Are p and q contradictory?Explain your answer.

1. p: EF 5 FG q: EG 5 2EF

No; F could be the midpoint of EG.

2. p: circumference of (D 5 2πx q: area of (D 5 4πx2

Yes; from p, the radius of (D is x; fromq, the radius of (D is 2x; for x > 0, x Þ 2x.


3. Write an indirect proof to show that B is not thereflection image of A over line ,.

Sample: If B is the reflection imageof A over line <, then by definitionAB ⊥ <. The ns formed would then containtwo right angles, which is impossible. Hence,B is not the reflection image of A over <.4. Write an indirect proof to show that Þ .

Sample: If 5 , then ( )25 ( )2.So 35 .

But ≈ 3.00032. Therefore, Þ .Uses Objective H

5. A softball team needs to pick a captain and an assistant captain. The captain must be either a junior or a senior. Tonya, Jodi, Jennifer, and Sarah all would like to be either captain or assistant captain. Jodi and Jennifer are the only sophomores. There are no freshmen on the team. Sarah will not serve unless Jodi also serves and vice versa. Use an indirect proof to show that Tonya cannot be the assistant captain.

Sample: Suppose Tonya is assistant captain.Then Sarah must be captain, since Jodi andJennifer cannot serve. But, Sarah will serveonly if Jodi serves. Thus, Tonya cannot beassistant captain.

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1. If G 5 (6, 3), E 5 (4, -1), R 5 (2, 7), andM 5 (-2, 3), use an indirect proof to

show that ←→GE is not parallel to

←→MR.

Sample: Suppose←→GE //

←→MR.

Then slope←→GE 5 slope

←→MR.

Slope ←→GE 5 5 2 and

slope ←→MR 5 5 1. Since

2 ≠ 1,←→GE is not parallel to

←→MR.


2. Prove that the quadrilateral with vertices P 5 (19, 13),Q 5 (15, 10), R 5 (18, 6), and S 5 (22, 9) is a rectangle.

Sample: Slope of 5 5 , of 5

5 - , of 5 5 , of 5 5 - .

So ⊥ and , and ⊥ and .

Thus PQRS is a rectangle.3. Prove that nEMG with vertices E 5 (-1, 9), M 5 (6, 7),

and G 5 (4, 0) is a right triangle.

Sample: Slope of 5 5 - , of 5

5 . So ⊥ and nEMG is a right triangle.

Representations Objective KIn 4–6, draw the figure in a convenient locationon the coordinate system.

4. an isosceles right triangle 5. a parallelogram 6. a square

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A

(0, 0)

(0, a)

(a, 0)

y

x

(-c, b) (a, b)

(-a, -b) (c, -b)

y

x

(a, a)

(0, 0)

(0, a)

(a, 0)

y

x

2

-2-2 2 6

6

y

xE = (4, -1)

G = (6, 3)

R = (2, 7)

M = (-2, 3)

Samples aregiven.

Name


Skills Objective AIn 1 and 2, find the distance between the given points.

1. (4, -3) and (-2, 6)

≈ 10.8

2. (1.5, 3.9) and (-0.4, -9.2)

≈ 13.2

3. What is the radius of the circleat the right with center D?

≈ 6.4


4. Use the figure at the right. Write anindirect proof to show that ABCD isnot a kite.

Sample: If ABCD is akite, then AD 5 CD or AB.But AD 5 5 ,CD 5 5 , and AB 5 5 .Therefore, ABCD is not a kite.Properties Objective G

5. nXYZ has vertices X 5 (85, 150), Y 5 (115, 110),and Z 5 (125, 180). Prove that nXYZ is isosceles.

Sample: XY 5 5 ; YZ 5

5 ; ZX 5 5

. So XY 5 ZX, and nXYZ is isosceles.Uses Objective I

6. Gary Ozawa gave you the following directions to get fromschool to his home: Go six blocks east, then 15 blocks north,then 4 more blocks east, then 1 block south, and finally1 block west. By air to his home, about how many kilometersis it, if 5 blocks is about 1 kilometer? ≈ 3.3 km

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D (5, 6)

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C (0, 2)

D (-3, 0)

A (-2, 4)

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Representations Objective JIn 1–5, write an equation for the circle satisfyingthe given conditions.

1. radius 6, center (-5, 2) 2. radius 14, center at the origin

(x15)21(y22)2536 x 2 1 y2 5 196

3. diameter 20, center (45, 57) 4. radius 2.5, center (h, k)

(x245)21(y257)25100 (x2h)21(y2k)256.25

5. center (4, -10), containing point (12, -10) (x24)21(y110)2564

6. a. On the grid at the right,draw the circle with radius 7and center at (0, 4).

b. Name 2 points on the circle.

Samples: (7,4), (0, 11)

c. Write an equation forthis circle.

x2 1 (y 2 4)2 5 49

In 7–10, an equation for a circle is given.a. Give its center. b. Give its radius.c. Name two points on the circle.

7. (x 2 8)2 1 ( y 1 3)2 5 169 8. x 2 1 (y 2 5)2 5 272.25

a. (8, -3) a. (0, 5)

b. 13 b. 16.5

c. Sample: (8, 10), c. Sample: (16.5, 5),(21, -3) (-16.5, 5)

9. (x 2 15)2 1 ( y 2 9)2 5 60 10. 86 5 x 2 1 y 2

a. (15, 9) a. (0, 0)

b. ≈ 7.7 b. ≈ 9.3

c. Sample: (15, 91 ), c.Sample: ( , 0), (15 1 , 9) (0, - )ÏÏ8866ÏÏ6600

ÏÏ8866ÏÏ6600

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A

(0, 4)

y

x5-5

5

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Skills Objective AIn 1 and 2, determine the coordinatesof the midpoint of the given segment.

1. a segment with endpoints (7,-4) and (-1, 9)

(3, ), or (3, 2.5)

2. a segment RS at the right

( , 2), or (2.5, 2)

3. Find the midpoint of MZ, given thatZ 5 (5, 1) and that M is the midpoint of thesegment with endpoints (14, -3) and (28, 17). (13, 4)

Skills Objective B

4. In nEFG at the right, A and B are midpoints of EG andFG, respectively.

a. If EF 5 12 cm, AB 5 6 cm .

b. What other relationship exists between EF and AB?

//5. G, H, and I are midpoints of the sides of nJKL as

shown at the right. If GI 5 7 cm and JL 5 26 cm, give all other segment lengths that can be found.

JK 5 14; JH 5 HK 5 7;

JI 5 IL 5 HG 5 13


6. Prove that in parallelogramWXYZ, the midpoints of thediagonals coincide.

Sample: Midpoint of

5 , 5 , 5 ;

midpoint of 5

, 5 , 5 .

Thus, , 5 is the midpoint of both and .WWYYXXZZ))7722((

))7722(())22 11 88

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W (–3, 2) Z (7, 2)

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Skills Objective C

1. a. Plot A 5 (4, -2, 1) and B 5 (-2, 5, 3)on the grid at the right.

b. Give the coordinates of the midpointof the segment joining A and B.

(1, , 2)

c. Find the distance between A and B.

≈ 9.4

Uses Objective I

2. A small closet is 160 cm tall, 80 cm deep, and 100 cm wide.Will a pair of skis which are 200 cm long fit in the closet?Explain your answer.

Sample: Yes; the diagonal of the closet is

≈ 205 cm.

Representations Objective J

3. A sphere has equation (x 1 3)2 1 ( y 2 6)2 1 ( z 1 2)2 5 289.

a. What is the center of the sphere? (-3, 6, -2)

b. What is its radius? 17

c. Give the coordinates of two pointsSample:

on the sphere. (-3, 6, 15), (14, 6, -2)

4. Write an equation for the sphere graphed at the right with center (0, 0, 3).

x2 1 y2 1 (z 2 3)2 5 36

ÏÏ11660022 11 880022 11 11000022

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A

x

B

A

z

y

(0, 0, -3)

(0, 0, 9)

z

y

x

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Representations Objective GIn 1–3, quadrilateral WXYZ is graphed at the right. W 5 (-4, -6),X 5 (-5, 4), Y 5 (2, 6), and Z 5 (5, 1).

1. List the coordinates of the vertices ofthe image of WXYZ under S1.5 andgraph it.

W’ 5 (-6, -9), X’ 5 (-7.5, 6),Y’ 5 (3, 9), Z’ 5 (7.5, 1.5)

2. List the coordinates of the vertices of S (WXYZ) and graph it.

W’’ 5 (-3, -4.5), X’’ 5 (-3.75, 3),Y’’ 5 (1.5, 4.5), Z’’ 5 (3.75, .75)

3. Give the coordinates of the image of Xunder Sk.

X’ 5 (-5k, 4k)

In 4–6, Let M 5 (10, 2), N 5 (7, 12), and O 5 (-1, 6).Let S4(nMNO) 5 nM’N’O’.

4. M’ 5 (40, 8) N’ 5 (28, 48) O’ 5 (-4, 24)

5. Verify that the slope of MN equals the slope of the line through M’ and N’.

slope of MN 5 5 ; slope of M’N’ 5

5 5

6. Use the distance Formula to verify that N’O’ 5 4 NO.

NO 5

N’O’ 5 5

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Skills Objective A

1. Draw the image of ABCD under a size change with center E and magnitude 2.

2. Draw the image of nJKL under a size change with center Kand magnitude 0.7.


3. For the figure at the right, use aruler to determine the center Cand scale factor k for the sizetransformation represented. The image is shown by the dashed line.

k 5

4. In the figure at the right, X’Y’Z’ is asize-change image of XYZ.

a. Is this size change anexpansion or a contraction?

expansionb. If X’Y’ = 10 and XY = 8, find

the magnitude k of the size change.

k 5 1.25c. Use the value of k in Part b to find YZ if Y’Z’ 5 6.4. YZ 5 5.12

3322

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K = K’L’

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Skills Objective AIn 1 and 2, draw the image of the figure underthe size change with center C and magnitude k.

1. k 5 2 2. k 5 .6


3. An architect’s original sketch of a building has length.8 m and height .5 m. The design is changed so that onthe sketch the new length is .32 m.

a. What is the scale factor of the contraction? k 5 .4

b. What is the new height of the building? .2 m

c. Find the area of the front face of each sketch of the building.

original .4 m2new .064 m2

____________________ ____________________

d. The area of the original sketch of the building is how many times the area of the new sketch of the building? 6.25 times

4. At the right, nLMN is a size-transformationimage of nIJH with center O. If OH 5 10,JH 5 7, ON 5 25, and m ∠ LMN 5 32, find

a. the magnitude ofthe size change. k 5 2.5

b. m∠ HJI. 32

c. MN. 17.5O H

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Skills Objective B

In 1 and 2, nBCD is the image of nEFGunder a size change.

1. If BC 5 2.8, EF 5 5.6, and CD 5 5.2,find FG.

FG 5 10.4

2. Multiple choice. Whichequation is a proportion? c

(a) (b)

(c) (d)

In 3 and 4, VIDEO is the image of ACTUL under a size change.

3. If DE 5 12, EO 5 9, and UL 5 15,find TU.

TU 5 20

4. If VO 5 12, AL 5 20, and CT 5 11,find the length of another segment.

ID 5 6.6

Uses Objective E

5. If 100 sheets of paper cost $1.19, about how muchwill 250 sheets cost? ≈ $2.98

6. If p pounds of carrots cost d dollars, at that rate,how much will q pounds of carrots cost?

dollars

7. One law of physics states that the amount of pressure that a gas exerts is proportional to the temperature of the gas in Kelvins (K), assuming a constant volume. If a gas exerts 16 lb/in2 of pressure at a temperature of 298 K, how much pressure would the gas exert at 278 K? ≈ 14.9 lb/in2

dqp

BCCD 5

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BDEF 5

BCEG

BCEF 5

BDFG


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Skills Objective B

1. TILE ; WALK, with sides and measuresas indicated. Find as many missinglengths and angle measures as possible.

m∠ K 5 55; m∠ I 5 112;AW ≈ 6.1; LE ≈ 7.4

In 2 and 3, nCBA , nONM. The ratio ofsimilitude is 2.35.

2. If NM 5 7.8, give a length of nCBA. BA ≈ 3.3

3. If m ∠ C 5 59, find the measure of another angle. m∠ O 5 59

4. In the figure below, rectangle EFGH is similar torectangle MJKL, JK 5 16, and EF 5 11.

a. What is a ratio of similitude?≈ .69 or ≈ 1.45

b. If EH 5 6, find MJ. ≈ 8.7

Uses Objective E

6. A Chinese tapestry is 16 feet high and 28 feet long.A reproduction of the original is 10 feet long. Howhigh is the reproduction? ≈ 5.7 ft

7. A model car is similar to its original. The model is20 cm long and has wheels which are 2 cm in diameter. If the original car is 5 m long, what is the diameter of the actual wheel? 50 cm, or .5 m

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1. Suppose two figures are similar and their lengths are inthe ratio 7 to 1. What is the ratio of their volumes? 343 to 1

2. Two similar isosceles trapezoids have areas of 40 and120 square units. If the longer base of the larger trapezoidhas a measure of 14 units, what is the measure of thecorresponding base of the smaller trapezoid?

≈ 8.1 units

3. Two triangular prisms are similar with ratio ofsimilitude 0.8. If the larger prism has volume250 cubic centimeters, find the volume of theother prism. 128 cm3

4. Two similar rectangles have perimeters of 76 ft and114 ft, respectively. What is the ratio of their areas?

or

Uses Objective F

5. The model of a cylindrical satellite is similar to an actualsatellite. If the model has a base area of 1.65 squaremeters and the model is the actual size of the satellite,

what is the base area of the actual satellite? 165 m2

6. The floor areas of two similar gymnasiums are14,400 square feet and 6400 square feet. If the lengthof the floor in the smaller gymnasium is 128 feet, what isthe length of the floor in the larger gymnasium? 192 ft

7. Two similar teddy bears’ heights are 50 cm and 20 cm.What is the ratio of their volumes?

or

8. Stop signs are shaped like regular octagons. A regulationstop sign has sides 32 cm long, while the stop sign for a snowmobile trail has sides 8 cm long. What is the ratioof the areas of the two stop signs?

or 116

161

8125

1258

110

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Uses Objective F

1. Two similar ancient Egyptian vases are 1 ft and 2 ft tall.If it takes 1 pint of paint to restore the larger vase, howmuch paint would the smaller vase require?

pint

2. A bakery sells bread based on total volume. A loaf whichis 50 cm long sells for $1.69. What is the cost of a similarloaf of bread 30 cm long? ≈ $.37

3. The model of a 350-foot-tall building is 2 feet tall.

a. What is the ratio of the base areas of the building andthe model?

b. If the area of the base of the model is 20 square inches,what is the area of the base of the actual building? ≈ 4253 ft2

4. A class wants to have a giant pizza for a party. They estimatethey will need a pizza with a 32-inch diameter. Suppose thelocal pizzeria agreed to make this giant pizza. If a 14-inch pizza costs $12.99, how much should the class expect topay for a similar 32-inch pizza? $67.87

5. Marta and Maggie do yard work to make money.They tell their employers that they charge $10 forevery 300 square feet of lawn. A new neighbor has a corner lot pictured at the right. Marta thinks they willearn $125, but Maggie says they will earn only $10. Who is correct? Justify your answer.

Marta; the area of the lawnis 3750 ft2 yielding a charge

of $125.

75 ft

100 ft125 ft

122,5004

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Properties Objective FIn 1–4, determine whether or not the triangles are similar. If they are similar, write a similarity statement using the correct order of vertices. Justify your answer.

1. 2.

nHIJ , nKLM; nPRA ,, nUBL;

3. nEFG: EF 5 , FG 5 , GE 5

nBCD: BC 5 12, CD 5 35, DB 5 42

nEFG ï nBCD; 5 5 , 5 Þ

4.nTOP: TO 5 28, OP 5 45, PT 5 56nBAT: BA 5 81, AT 5 100.8, TB 5 50.4

nTOP ,, nTBA; 5 5 5 1.8

In 5 and 6, the triangles are similar with corresponding sides parallel. a. Find the ratio of similitude. b. Write a similarity statement using the correct order of vertices.

5. 6.

a.k 5 or

a.k 5 or 2

b. nBMN ,, nBAC b. nPOQ ,, nRSQ

12

75

57

100.856

8145

50.428

301

632

424/ 3

301

357/6

122/5

43

76

25

64 5 4.5

3 5 24/3 5 3

27.75.5 5 8.4

6 5 9.16.5 5 7

5

4

343

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Samples aregiven.

25

410

15

6

A

M

B N C

4.8

2.9

3.5

5.8 PO

R SQ

Samples aregiven.

Name


Properties Objective FIn 1–4, determine whether the triangles are similar.If so, what triangle similarity theorem guaranteestheir similarity?

1. 2.

yes; AA Simil. Thm.__ yes; SSS Simil. Thm.-

3. 4.

yes; SAS Simil. Thm.__ yes; SAS Simil. Thm.

5. In the figure at the right, ←→XY //

←→ST,

Prove that nSTC , nYXC.

Sample argument: Since ←→XY //

←→ST, ∠ CST > ∠ CYX

because // lines ⇒ AIA >.∠ TCS > ∠ XCY by the VerticalAngles Theorem. Hence, by the AASimilarity Theorem, nSTC , nYXC.Uses Objective H

6. A tree casts a shadow that is 8 m long atthe same time a meter stick casts a shadow40 cm long. How tall is the tree? 2000 cm or 20 m

7. A freeway ramp is 10 ft high after 200 ft.How high is the highest point of the ramp ifthe ramp is 1500 ft in length? 75 ft

24

13.2

2

1.252

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Skills Objective A

1. In nABC, // . 2. In nJKL, // .Find CF. Find JX.

CF 5 5 JX ≈ 1.4

In 3 and 4, use the figure at theright, in which PQ // ON, MP 5 10, and PO 5 45.

3. If MN 5 100, what is MQ?MQ 5 18

4. If OR 5 66, what would RN haveto be so that PR is parallel to MN?

RN 5 14

5. Use the figure below. Name allpairs of parallel lines and explain whythey are parallel. The figure is notnecessarily drawn accurately.

5 , so ←→WX //

←→VU by the Side-Splitting

Converse Theorem.

Uses Objective HIn 6 and 7, use the drawing of the roof below.

6. Given that TO 5 6 m, OP 5 4 m,TR 5 7.2 m, and RE 5 4.8 m, areOR and PE parallel?

yes7. If PE 5 12 m, what is the length of

the support OR?

7.2

915

35

23

211

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4.8

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M

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4 m

12 m

4.8 m

7.2 m

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Vocabulary

1. If a, b, and g . 0, what proportion guaranteesthat g is the geometric mean of a and b?

5

Skills Objective AIn 2–4, Use n MNO below at the right.

2. If ML 5 5 and LN 5 20, then

OL 5 10 .

3. If ML 5 8 and OL 5 16, then

MN 5 40 .

4. If ON 5 15 and MN 5 17, then

LN 513

.

5. What is the length of BD in nABCat the right?

BD 5

6. MK is the altitude to the hypotenuse ofn AME at the right. AE 5 10 andKE 5 9, find MA, ME, and AK.

MA ME 3 AK 1

7. In the 3-4-5 right triangle TFV at the right,find the length of GF, GT, GV, and GH.

GF 2.4 GT 1.8

GV 3.2 GH 1.92

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Skills Objective C

1. In nPQR at the right, find PQ and QR.

PQ 14

QR 7 < 12.1

2. In nKLM at the right, find KL and KM.

KL 4 ≈ 5.7

KM 4 ≈ 5.7

3. An isosceles triangle has base angles of 30°. Thelength of the base is 14. What is the length of thealtitude to the base? ≈ 4.04

4. In the diagram at the right, if JA 5 10and IE 5 12, find each length.

JI 12

AM 10

ME 2 ≈ 2.8

JM 10 ≈ 14.1

5. In nGSM at the right, GN 5 18 and N is themidpoint of SM. Find each length.

SN 6 ≈ 10.4

GS 12 ≈ 20.8

NM 6 ≈ 10.4

GM 12 ≈ 20.8Ï3Ï3Ï3

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Skills Objective D

1. Draw a right triangle with legs measuring8 units and 15 units. Use this triangle toestimate ∠ G if tan G 5 .

m∠ G ≈ 28

2. Use nAMC at the right. Find each ratio.

a. tan M

b. tan C

Skills Objective E

3. In nJES, use a protractor to determinem∠ E to the nearest degree, and thencalculate tan E to the nearest hundredth.

m∠ E ≈ 25, tan E 5 .47

Properties Objective GIn 4 and 5, use nQRS at the right. Do not measure.

4. What is the tangent of ∠ Q?

5. is the tangent of

which angle? ∠ S

Uses Objective I

6. Gertie the gopher is looking at a tree that is 100 feetaway from her hole. If the angle of elevation betweenGertie’s hole and the top of the tree is 23°, how tall isthe tree? 42.4 ft

QRRS

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Skills Objective DIn 1–3, use n JKL at the right. Find the indicated ratio.

1. sin K .6

2. cos K .8

3. cos J .6

Skills Objective E

4. Estimate sin 68° to the nearest hundredth. .93

In 5 and 6, give exact values.

5. cos 45° 6. sin 60°


7. Define cos A.

cos A 5

In 8 and 9, use nHIL at the right.

8. is the __sine___ of angle H.

9. Write a ratio for cos H.

Uses Objective I

10. What is the measure of the angle x madeby a 200-ft supporting cable with a 150-ft-tallradio tower?

41.4

11. How far up on the side of a building can a 15-m ladder reach if the measure of the angle it makes with the ground may not exceed 72? 14.3 m

HLHI

ILHI

leg adjacent to ∠Ahypotenuse

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Skills Objective DIn 1–4, determine the area of the triangle.

1. 2.

≈ 22.7 units2 ≈ 563.7 units2

3. 4.

≈ 12.3 units2 ≈ 28.2 units2

Uses Objective 1

5. A cruise ship travels at 18 kilometers per hour in acourse 54° south of west. Find the southern andwestern exponents of its velocity.

S ≈ 14.6 km/h W ≈ 10.6 km/h

6. A hiker needs to reach a point 3 miles north and12 miles east of her present location. 76° E of N ora. In what direction must she hike? 14° N of E

b. How far will she need to travel? ≈ 12.4 miles

7. A submarine traveled 450 miles in the direction25° north of west. It then traveled 210 milesin the direction 74° north of west. To thenearest mile, how far has the submarinetraveled from its starting point? ≈ 609 miles

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Skills Objective AIn 1–3, the circle has a radius of 6 cm. Findthe length of the chord of the indicated arc.

1. a 90° arc 6 ≈ 8.49 cm

2. a 60° arc 6 cm

3. a 132° arc ≈ 10.96 cm

In 4 and 5, use nMJK in (K with radius 3.5 in.

4. If m∠ JKM 5 84, then mCMJ 5 ______84°______ .

5. Find MJ and MK.

MJ ___4.68 in.___ MK ____3.5 in.____

6. True or false. If nYES is equilateral,then m CYE 5 mCSE 5 120°.

true

Properties Objective FIn 7–9, square QUAD is inscribed in (C,and < bisects ∠ QCU. Justify the statement.

7. QB 5 BU Samples are given.Chord-Center Thm. (3)

8. mCDA 5 m

CAU

Arc-Chord Congruence Thm. (2)9. , // QD

Chord-Center Thm. (3) and Two Perpendiculars Thm.

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Skills Objective BIn 1–4, use the circle at the right. Find theindicated measure.

1. mCWXZ 198°

2. m∠ WXZ 81

3. m CXZ 134°

4. m∠ YZW 58

Skills Objective C

In 5–9, use (R at the right, in which BUis a diameter, m∠ U 5 37, and m CBE 5 135°.Find the indicated measure.

5. mCBL 74°

6. m CEU 45°

7. m∠ ELU 22.5

8. m∠ LTU 120.5

9. m∠ BTE 120.5

Properties Objective GIn 10 and 11, use ( Q at the right.

10. Explain why nRST is a right triangle.

Sample: ∠ T is a right angle

because it is inscribed in asemicircle. By definition, nRST is a right triangle.

11. If m∠ R 5 x, then mTRC5

___(180 2 2x)___ .


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Skills Objective DIn 1 and 2, use the right-angle method to Samples are given.find the center O of the circle.

1. 2.

Uses Objective IIn 5–7, use the diagram at the right of the space shuttle. The actual shuttle is 184 feet long.

5. Locate all points where a photographer could standto fit CA exactly in the picture if the camera lens has a picture angle with measure of 65.

6. a. How far from point M, the midpoint of CA, willthe photographer have to be if she stands on theperpendicular bisector of CA? ≈144.4 ft

b. Mark this point x on the diagram.

7. How far would a different photographer need to be from M if his camera lens angle measures 56 andhe stands on the perpendicular bisector of CA? ≈ 173.0 ft

8. In general, the ....?.... the picture angle, the greater the distance one must stand to fit an object exactly into a picture. smaller

4. Draw a circle through thethree points C, A, and T.

3. Draw the entire circle of thecircular arc below. Thenfind the center of the circle.

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Skills Objective CIn 1–3, use the circle at the right.

1. If m∠ GHI 5 52 and m CGI 5 44°, what other arcmeasure can be determined? What is its measure?

mCFJ 5 60°2. If m CFG 5 130° and m CIJ 5 120°,

what is m∠ IHJ? _-__125_-__

3. If m CFG 5 133° and mCIJ 5 117,°what is m ∠ GHI? __-__55__-__

In 4–6, use (Q at the right. Assume thatmCNO 5 30° and mCRS 5 70°.

4. What is m ∠ M? __,__20__,__

5. What is m ∠ RPS? __,__50__,__

6. List all other angle measures that you can.

m∠ RNS 5 m∠ ROS 5 35; m∠ NSO 5 m∠ NRO 5 15;m∠ NPR 5 m∠ OPS 5 130; m∠ NPO 5 50


7. Write an argument to complete the proof.Given: (A with segments and

measures as given at the right.To prove: nOYC is an isosceles right triangle.

Sample argument: By the Angle-Chord Thm., m∠ OYC 5

(165 1 15) 5 90. Since m COR

5 m CCK 5 x°, OR > CK by the Arc-Chord >Thm. Then, m∠ ROK 5 15 and m∠ KCR 5

15, so ∠ ROK > ∠ KCR. By the Vertical AnglesThm., ∠ OYR > ∠ CYK, and nORY > nCKY by the AAS > Thm. So, OY > CY and nOYC is an isosceles right triangle by definition.

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In 1–3, ←→GS and

←→GN are tangents to

( M at S and N.

1. a. What is the measure of ∠ GSM?

90b. What kind of figure is GSMN?

kite

2. If SM 5 8 and GS 5 15, find the area of GSMN. 120 units2

3. If m∠ SMN 5 174, what is m CSE? 87°


Given: m∠ OPQ 5 90; OP // ,.To prove: , is tangent to (P at point Q.

Sample argument:

Uses Objective JIn 5–7, assume the radius of the earth is3960 miles or 6375 kilometers and that thereare no hills or obstructions.

5. If a person could stand on the tip of the torch of theStatue of Liberty, 91.5 m above the water, how farcould the person see? ≈ 34.2 km

6. The most-recently accepted height of Mt. Everest is29,028 ft. How far could a person see when standing on its summit? ≈ 208.7 mi

7. To the nearest foot, how far above the earth is a planeif the pilot can see 8 miles? 43 ft

Justificationsdefinition of ⊥Perpendicular toParallels Thm.Radius-Tangent Thm.

Conclusions1. OP ⊥ PQ2. PQ ⊥ <

3. < is tangentto (P at Q.


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Skills Objective CIn 1–4, use the figure at the right, with P the center of the circle. Find the indicated measure.

1. m∠ JIK 12

2. mONC 45°

3. mLMC 56°

4. m∠ JIM 45

In 5–7, use (P at the right. Find theindicated angle measure if mTAC 5 35°.

5. m∠ LSE 55

6. m∠ SLE 90

7. m∠ SAL 107.5


8. Given that XY and XZ are tangent to (U at points Y and Z, if m

CYWZ = 236° list all of the

angle measures and arc measures that you can.

m∠ XYU 5 m∠ XZU 5 90; m∠ YUX 5 m∠ ZUX 562; m∠ YXU 5 m∠ ZXU 5 28; m∠ YUZ 5 124;m∠ YXZ 5 56; mYZC 5 124°Properties Objective G

9. Write an argument to complete the proof. Sample:Given:

←→WV is tangent to (T at V,

and ←→WV // SU.

To prove: nSVU is isosceles.


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1. m∠ S 5 m∠ SVW2. m∠ SVW 5 mCSV

3. m∠ U 5 mCSV4. m∠ S 5 m∠ U5. nSVU is isos.

// lines ⇒ AIA >Tangent-Chord Thm.

Inscribed Angle Thm.Transitivity and Sub.def. of isos. n

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Skills Objective E

1. L, M, N, and O all lie on (Q at the right. IfLP 5 33, PN 5 2, and MP 5 3, find OP.

22________________

In 2 and 3, use (R at the right.

2. If EA 5 18 and AL 5 2, find LH.

2 ≈ 6.33. If LH 5 10 and EA 5 25, find AL.

≈ 3.5________________

In 4 and 5, use (M at the right.

4. If AK 5 4, KS 5 10, and AN 5 5, find PN.

6.2________________

5. If AS 5 16, AK 5 6, and AP 5 12,

a. find AN.

8________________

b. find PN.

4________________

6. Use (C at the right. ←→AB is tangent

to (C at B.

a. What is the power of point A?

AB2________________

b. If AJ 5 18 and JK 5 60, find AB.

≈ 37.5________________

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1. Consider all figures with area 40 square centimeters.

a. Which has the least perimeter? circle

b. What is that perimeter? ≈ 22.4 cm

2. A circle and a regular polygon each have a perimeterof 10 inches. Which has less area? reg. polygon

3. Of all triangles with a fixed perimeter, which has the greatest area? equilateral

triangle4. A rectangle has sides measuring 2.5 mm and 3.8 mm.

What is the greatest possible area of a figure withthe same perimeter? 12.6 mm2

Uses Objective K

5. a. Draw a figure with a large b. Draw a figure with a largearea for its perimeter. perimeter for its area.

Sample: Sample:

6. Northfield wants to build a park with an area of about8000 square meters.

a. What would be the least perimeter possiblefor the park? ≈ 317 m

b. If a park with a quadrilateral shape is desired, whatwould be the least perimeter? ≈ 358 m

7. A fence encloses a grazing area as shownat the right.

a. Find the area of the region.

112,500 ft2________________

b. Find the area of the largest region thatcould be enclosed by this fence.

≈ 183,856 ft2________________


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1. Consider all figures in space with a surface areaof 100 square feet.

a. Which has the greatest volume? sphere

b. What is this volume? ≈ 94.0 ft3

2. A sphere and a square-based pyramid each havea volume of 2400 cm3. Which has the greatersurface area? pyramid

3. Of all regular pyramids or cones with equalsurface area, which has the greatest volume? cone

4. Of all the rectangular prisms with a fixedvolume, which has the least surface area? cube

Uses Objective K

5. a. Draw an object with a large b. Draw an object with a largevolume for its surface area. surface area for its volume.

Sample: Sample:

6. A container is to be designed for 36 in3 of butter. Find its surface area if the container is

a. a cube. ≈ 65.4 in2

b. a sphere. ≈ 52.7 in2

c. a right cone with height 4 inches. ≈ 72.7 in2

7. Give two reasons why the cube would be the bestcontainer in Question 6.

Samples: It stacks efficiently; its surfacecan be printed upon accurately.

8. A jeweler has enough gold to cover a surface area of 3 cm2

with a particular thickness of gold.

a. What is the volume of the largest bead she could cover? ≈ .489 cm3

b. What is the volume of the largest prism-shaped beadshe could cover? ≈ .354 cm3

lesson masters a -...

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