foundations of geometry - advisory...

7A Two-DimensionalGeometry

7-1 Points, Lines, Planes,and Angles

LAB Bisect Figures

7-2 Parallel and Perpendicular Lines

LAB Constructions

7-3 Angles in Triangles

7-4 Classifying Polygons

LAB Exterior Angles of aPolygon

7-5 Coordinate Geometry

7B Patterns in Geometry7-6 Congruence

7-7 Transformations

LAB Combine Transformations

7-8 Symmetry

7-9 Tessellations

PlaygroundEquipment Designer

Playground equipmentmust be attractive, safe, fun,and appropriate for the agesof the children who will useit. Years ago, designers usedpencils, T-squares, and sliderules to create their designs.Designers now use computers,3-D programs, and virtual reality to design playgrounds.

Foundations ofGeometryFoundations ofGeometry

320 Chapter 7

KEYWORD: MT7 Ch7

Shapes of Playground Equipment

EquipmentGround Shape

Merry-go-round Circle

Four-square court Square

SwingsRectangle

Climbing structure Octagon

m807_c07_320_321 1/13/06 11:45 AM Page 320

Foundations of Geometry 321

VocabularyChoose the best term from the list to complete each sentence.

1. In the __?__ (4, �3), 4 is the __?__, and �3 is the __?__.

2. The __?__ divide the __?__ into four sections.

3. The point (0, 0) is called the __?__.

4. The point (0, �3) lies on the __?__, while the point (�2, 0) lies on the __?__.

Complete these exercises to review skills you will need for this chapter.

Ordered Pairs Write the coordinates of the indicated points.

5. point A 6. point B

7. point C 8. point D

9. point E 10. point F

11. point G 12. point H

Similar FiguresTell whether the figures in each pair appear to be similar.

13. 14.

EquationsSolve each equation.

15. 2p � 18 16. 7 � h � 21 17. �3x

� � 9 18. y � 6 � 16

19. 4d � 1 � 13 20. �2q � 3 � 3 21. 4(z � 1) � 16 22. x � 3 � 4x � 23

Determine whether the given values are solutions of the given equations.

23. �23�x � 1 � 7 x � 9 24. 2x � 4 � 6 x � �1

25. 8 � 2x � �4 x � 5 26. �12�x � 5 � �2 x � �14

O 4

4

2

2�2�4

�2

�4

x

y

A

B

CD E

F

G H

coordinate axes

coordinate plane

ordered pair

origin

x-axis

x-coordinate

y-axis

y-coordinate

m807_c07_320_321 1/13/06 11:45 AM Page 321

Previously, you

• located and named points on a coordinate plane.

• recognized geometric conceptsand properties in fields such asart and architecture.

• used critical attributes to define similarity.

You will study

• graphing translations andreflections on a coordinateplane.

• using geometric concepts andproperties of geometry to solveproblems in fields such as artand architecture.

• using critical attributes todefine congruency.

You can use the skillslearned in this chapter

• to find angle measures by usingrelationships within figures.

• to create tessellations.

• to identify properties ofgeometry in art andarchitecture.

Vocabulary ConnectionsTo become familiar with some of thevocabulary terms in the chapter, consider thefollowing. You may refer to the chapter, theglossary, or a dictionary if you like.

1. The word equilateral contains the rootsequi, which means “equal,” and lateral,which means “of the side.” What do yousuppose an is?

2. The Greek prefix poly means “many,” andthe root gon means “angle.” What do yousuppose a is?

3. Think of what means when you aretalking about a hill. How do you think thisapplies to lines on a coordinate plane?

slope

polygon

equilateral triangle

322 Chapter 7 Foundations of Geometry

Key Vocabulary/Vocabularioequilateral triangle triángulo equilátero

line línea

parallel lines líneas paralelas

perpendicular lines rectas perpendiculars

plane plano

point punto

polygon polígono

reflection reflexión

slope pendiente

transformation transformación

translation translación

transversal transversal

Stu

dy

Gu

ide:

Pre

view

m807_c07_322 1/13/06 11:45 AM 22

January 27

I’m having trouble with Lesson 6-5. I can find what

percent one number is of another number, but I get

confused about finding percent increase and decrease.

My teacher helped me think it through:

Find the percent increase or decrease from 20 to 25.

• First figure out if it is a percent increase or decrease.

It goes from a smaller to a larger number, so it is

a percent increase because the number is getting larger,

or increasing.

• Then find the amount of increase, or the difference,

between the two numbers. 25 � 20 � 5

• Now find what percent the amount of increase, or

difference, is of the original number.

�am

o

o

r

u

ig

n

i

t

na

o

l

f

n

i

u

n

m

cr

b

e

e

a

r

se� → �

2

5

0� � 0.25 � 25%

So it is a 25% increase.

Writing Strategy: Keep a Math JournalBy keeping a math journal, you can improve your writing and thinkingskills. Use your journal to summarize key ideas and vocabulary from eachlesson and to analyze any questions you may have about a concept oryour homework.

Journal Entry: Read the entry a student made in her journal.

Begin a math journal. Write in it each day this week, using theseideas as starters. Be sure to date and number each page.

■ In this lesson, I already know . . .

■ In this lesson, I am unsure about . . .

■ The skills I need to complete this lesson are . . .

■ The challenges I encountered were . . .

■ I handled these challenges by . . .

■ In this lesson, I enjoyed/did not enjoy . . .

Try This

Read

ing

and

Writin

g M

ath

Reading and Writing Math 323

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Learn to classify andname figures.

Vocabulary

vertical angles

congruent

supplementaryangles

complementaryangles

obtuse angle

acute angle

right angle

angleray

segmentplane

linepoint

Points, lines, and planes are the building blocks of geometry.Segments, rays, and angles are defined in terms of these basic figures.

GHG

H

A names a location. • A point A

A is perfectly straight and extends forever in both directions.

A is a perfectly flat surface thatextends forever in all directions.

A , or line segment, is the part of a line between two points.

A is part of a line that starts at one point and extends forever in one direction.

ray

segment

plane

line

point

Z

Q R S

T

J

K

KJ

BC�� is read “line BC.” GH�� is read “segment GH.” KJ�� is read “ray KJ.” Toname a ray, always write the endpoint first.

Naming Points, Lines, Planes, Segments, and Rays

Use the diagram to name each figure.

four points

Q, R, S, T

a line Any 2 points on the line can

Possible answers: QS��, QR�� or RS�� be used.

a plane

Possible answers: Any 3 points in the plane that plane Z or plane QRT form a triangle can name a plane.

four segments Write the 2 points in any order,

Possible answers: QR��, RS��, RT��, QS�� for example, Q�R� or R�Q�.

five rays

RQ��, RS��, RT��, SQ��, QS�� Write the endpoint first.

E X A M P L E 1

line �, or BC

BC

�

EPplane P, orplane DEFF

D


7-1 Points, Lines, Planes,and Angles

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A measures 90°. An measures greater than 0° and

less than 90°. An measures greater than 90° and less than

180°. are two angles whose measures add to 90°.

are two angles whose measures add to 180°.

Classifying Angles


a right angle two acute angles

�DEC �AED, �CEB

two obtuse angles

�AEC, �DEB m�AEC � 150°; m�DEB � 120°

a pair of complementary angles

�AED, �CEB m�AED � m�CEB � 60° � 30° � 90°

two pairs of supplementary angles

�AED, �DEB m�AED � m�DEB � 60° � 120° � 180°�AEC, �CEB m�AEC � m�CEB � 150° � 30° � 180°

figures have the same size and shape.

• Segments that have the same length are congruent.

• Angles that have the same measure are congruent.

• The symbol for congruence is �, which is read “is congruent to.”

Intersecting lines form two pairs of . Vertical angles arealways congruent, as shown in the next example.

vertical angles

Congruent

Supplementary angles

Complementary angles

obtuse angle

acute angleright angle

The measures of angles that fittogether to form a straight line,such as �FKG, �GKH, and�HKJ, add to 180°.

The measures of angles that fittogether to form a completecircle, such as �MRN, �NRP,�PRQ, and �QRM, add to 360°.

An (�) is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees. m�1 means the measure of �1. The angle can be named �XYZ, �ZYX, �1, or �Y. The vertex must be the middle letter.

angle

Y

1Z

X

F

GH

K JR

M

NP

Q

A

C

30°90°

60°

D

E BA right angle can belabeled with a smallbox at the vertex.

2E X A M P L E

7-1 Points, Lines, Planes, and Angles 325

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7-1 ExercisesExercises


1. three points 2. a line 3. a plane

4. three segments 5. three rays


6. a right angle 7. two acute angles

8. an obtuse angle 9. a pair of complementary angles

10. two pairs of supplementary angles

KEYWORD: MT7 Parent

KEYWORD: MT7 7-1

GUIDED PRACTICE

See Example 2

See Example 1

Think and Discuss

1. Tell which statements are correct if �X and �Y are congruent.

a. �X � �Y b. m�X � m�Y c. �X � �Y d. m�X � m�Y

2. Explain why vertical angles must always be congruent.

Finding the Measures of Vertical Angles

In the figure, �1 and �3 are vertical angles, and�2 and �4 are vertical angles.

If m�2 � 75�, find m�4.

m�1 � 180° � 75° �2 and �1 are supplementary.

� 105°

m�4 � 180° � 105° �1 and �4 are supplementary.

� 75°

So m�2 � m�4, or �2 � �4

If m�3 � x�, find m�1.

m�1 � 180° � m�4 �3 and �4 are supplementary.

m�1 � 180° � (180° � x°) Substitute 180° � x° for m�4.

� 180° � 180° � x° Distributive Property

� x° Simplify.

So m�1 � m�3, or �1 � �3.

2 43

1

E X A M P L E 3

AY

X

Z

EA

B C

D

50°40°

90°


m807_c07_324_328 1/19/06 4:05 PM Page 326

In the figure, �1 and �3 are vertical angles,and �2 and �4 are vertical angles.

11. If m�3 � 105°, find m�1.

12. If m�2 � x°, find m�4.


13. four points 14. two lines

15. a plane 16. three segments

17. five rays


18. a right angle

19. two acute angles

20. two obtuse angles

21. a pair of complementary angles

22. two pairs of supplementary angles

In the figure, �1 and �3 are vertical angles,and �2 and �4 are vertical angles.

23. If m�2 � 126°, find m�4.

24. If m�1 � b°, find m�3.

Use the figure for Exercises 25–34. Write true or false. If a statement is false,rewrite it so it is true.

25. NQ�� is a line in the figure.

26. Rays UQ�� and UT�� make up line TQ��.

27. �QUR is an obtuse angle.

28. �4 and �2 are supplementary.

29. �1 and �6 are supplementary.

30. �3 and �1 are complementary.

31. If m�1 � 35°, then m�6 � 40°.

32. If m�SUN � 150°, then m�SUR � 150°.

33. If m�1 � x° , then m�PUQ � 180° � x° .

34. m�1 � m�3 � m�5 � m�6 � 180°.

35. Critical Thinking Two complementary angles have a ratio of 1:2. What is the measure of each angle?

PRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICE

Extra PracticeSee page 794.

See Example 3

See Example 3

See Example 2

See Example 1

12

34

12

34

NJ

L

M

K

VWX

YZ

60°30°

Q

U

T

N

R

P

S

61

35

2

4

7-1 Points, Lines, Planes, and Angles 327

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Physical Science

40. Multiple Choice When two angles are complementary, what is the sum of their measures?

90° 180° 270° 360°

41. Gridded Response �1 and �3 are supplementary angles. If m�1 � 63°, find m�3.

Multiply. Write the product as one power. (Lesson 4-3)

42. m3 � m2 43. w � w6 44. 78 � 73 45. 116 � 119

46. Callie made a 5 in. tall by 7 in. wide postcard. A company would like to sell a poster based on the postcard. The poster will be 2 ft tall. How wide will the poster be? (Lesson 5-5)

DCBA

Physical ScienceThe archerfish can spit a stream of water up to 3 meters inthe air to knock its prey into the water. This job is mademore difficult by refraction, the bending of light waves asthey pass from one substance to another. When you lookat an object through water, the light between you and theobject is refracted. Refraction makes the object appear tobe in a different location. Despite refraction, the archerfishstill catches its prey.

36. Suppose that the measure of the angle between the bug’sactual location and the bug’s apparent location is 35°.

a. Refer to the diagram. Along the fish’s line of vision,what is the measure of the angle between the fish andthe bug’s apparent location?

b. What is the relationship of the angles in the diagram?

37. In the image, the underwater part of the net appears to be 40° to the right of where it actually is. What is themeasure of the angle formed by the image of theunderwater part of the net and the part of the net above the water?

38. Write About It Suppose an archerfish is directly below its prey.Explain why there would be little or no distortion.

39. Challenge A person on the shore is looking at a fish in the water. At the same time, the fish is looking at the person from below the surface.Describe what each observer sees, and where the person and the fishactually are in relation to where they appear to be.


m807_c07_324_328 1/13/06 11:46 AM Page 328

Follow the steps below to bisect a segment.

a. Draw J�K� on your paper. Place your compass point on Jand draw an arc. Without changing your compass opening, place your compass point on K and draw an arc.

b. Connect the intersections of the arcs with a line. MeasureJ�M� and K�M�. What do you notice?

Follow the steps below to bisect an angle.

a. Draw acute �H on your paper. b. Place your compass point on H and draw an arc through both sides of the angle.

c. Without changing your compass d. Draw HD��. Measure �GHD and �DHE.opening, draw intersecting arcs from What do you notice?G and E. Label the intersection D.

1. Explain how to use a compass and a straightedge to divide a segmentinto four congruent segments. Prove that the segments are congruent.

Draw each figure, and then use a compass and a straightedge to bisect it.Verify by measuring.

1. a 2-inch segment 2. a 0.5-inch segment 3. a 6-inch segment

4. a 48° angle 5. a 90° angle 6. a 110° angle

Bisect Figures

Use with Lesson 7-1

7-1

Think and Discuss

Try This

KEYWORD: MT7 Lab7When you bisect a figure, you divide it into two congruent parts.

J K

J KM

1

E

D

H

G

E

DH

G

H

EH

G

2

Activity

7-1 Hands-On Lab 329

m807_c07_329 1/13/06 11:46 AM Page 329

Learn to identifyparallel and perpendicular lines and the angles formed by a transversal.

Vocabulary

transversal

perpendicular lines

parallel lines

are lines in aplane that never meet, suchas the opposite edges of askyscraper’s windows. Theedges appear to get closerto each other because ofperspective.

An edge and the bottom of a window are like

; that is,they intersect at 90° angles.

A is a line that intersects two or more lines that lie in the sameplane. Transversals to parallel lines form angles with special properties.

Identifying Congruent Angles Formed by a Transversal

Copy and measure the anglesformed by the transversal andthe parallel lines. Which anglesseem to be congruent?

�1, �4, �5, and �8 all measure 60°.�2, �3, �6, and �7 all measure 120°.

Angles marked in blue appearcongruent to each other, andangles marked in red appearcongruent to each other. �1 � �4 � �5 � �8�2 � �3 � �6 � �7

transversal

perpendicular lines

Parallel lines

You cannot tell ifangles are congruentby measuring becausemeasurement is notexact.

The sides of the windows are transversals to the top and bottom.

The top andbottom of thewindowsare parallel.

1 23 4

5 67 8

1 23 4

5 67 8

E X A M P L E 1


7-2 Parallel and PerpendicularLines

m807_c07_330_333 1/13/06 11:46 AM Page 330

Finding Angle Measures of Parallel Lines Cut by Transversals

In the figure, line a⏐⏐line b. Find the measureof each angle.

�4

m�4 � 74� Corresponding angles are congruent.

�3

m�3 � 74° � 180� �3 is supplementary to the 74� angle.

� 74� � 74� Subtract 74� from both sides.�� m�3 � 106� Simplify.

�5

m�5 � 106� �3 and �5 are alternate interior angles, sothey are congruent.

Some pairs of the eight angles formed by two parallel lines and atransversal have special names.

PROPERTIES OF TRANSVERSALS TO PARALLEL LINES

If two parallel lines are intersected by a transversal, • corresponding angles are congruent, • alternate interior angles are congruent, • and alternate exterior angles are congruent.

If the transversal is perpendicular to the parallel lines, all of theangles formed are congruent 90� angles.

12

34

56

a

bc

7

74°

Alternate interior Alternate exterior Corresponding

Think and Discuss

1. Tell how many different angles would be formed by a transversalintersecting three parallel lines. How many different anglemeasures would there be?

2. Explain how a transversal could intersect two other lines so thatcorresponding angles are not congruent.

2E X A M P L E

The symbol forparallel is⏐⏐. Thesymbol forperpendicular is ⊥.

7-2 Parallel and Perpendicular Lines 331

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1. Measure the angles formed by thetransversal and the parallel lines.Which angles seem to becongruent?

In the figure, line m⏐⏐line n. Find themeasure of each angle.

2. �1 3. �4

4. �6 5. �7

6. Measure the angles formed by the transversal andthe parallel lines. Which angles seem to becongruent?

In the figure, line p⏐⏐line q. Find the measure of each angle.

7. �1

8. �4

9. �6

10. �7

In the figure, line t⏐⏐line s.

11. Name all angles congruent to �1.

12. Name all angles congruent to �2.

13. Name three pairs of supplementary angles.

14. Which line is the transversal?

15. If m�4 is 51�, what is m�2?

16. If m�7 is 116�, what is m�3?

17. If m�5 is 91�, what is m�2?

Draw a diagram to illustrate each of the following.

18. line p⏐⏐line q⏐⏐line r and line s transversal to lines p, q, and r

19. line m⏐⏐line n and transversal h with congruent angles �1 and �3

20. line h⏐⏐line j and transversal k with eight congruent angles

KEYWORD: MT7 Parent

KEYWORD: MT7 7-2

GUIDED PRACTICE




See Example 2

See Example 1

See Example 2

See Example 1

1 2 5 63 4 7 8

1

4 237

68

5

t

r

s

1 42

37

65

p q

110°

12

34

5 86

7

123

n

476

5

m

t

118°


m807_c07_330_333 1/13/06 11:46 AM Page 332

A

B

21. Critical Thinking Two parallel lines are cut by a transversal. Can youdetermine the measures of all the angles formed if given only one anglemeasure? Explain.

22. Physical Science A periscopecontains two parallel mirrors thatface each other. With a periscope, aperson in a submerged submarinecan see above the surface of thewater.

a. Name the transversal in the diagram.

b. If m�1 � 45°, find m�2, m�3, and m�4.

23. What’s the Error? Line a is parallel to line b. Line c is perpendicular toline b. Line c forms a 60° angle with line a. Why is this figure impossible todraw?

24. Write About It Choose an example of abstract art or architecture withparallel lines. Explain how parallel lines, transversals, or perpendicular linesare used in the composition.

25. Challenge In the figure, �1, �4, �6, and �7 are all congruent, and �2, �3, �5, and �8 are all congruent. Does this mean that line s is parallel to line t? Explain.

26. Multiple Choice Two parallel lines are intersected by a transversal. The measures of two corresponding angles that are formed are each 54°.What are the measures of each of the angles supplementary to thecorresponding angles?

36° 72° 108° 126°

27. Extended Response Suppose a transversal intersects two parallel lines.One angle that is formed is a right angle. What are the measures of theremaining angles? What is the relationship between the transversal andthe parallel lines?

Find each number. (Lesson 6-3)

28. What is 15% of 96? 29. What is 146% of 12,500?

30. What is 0.5% of 1000? 31. What is 99.9% of 1500?

�1 and �3 are vertical angles, and �2 and �4 are vertical angles. �1 and �2 aresupplementary angles. (Lesson 7-1)

32. If m�1 � 25°, find m�3°. 33. If m�2 � 95°, find m�3.

DCBA

Physical Science

This hat from the 1937 BritishIndustries Fair isequipped with a pair of parallelmirrors to enablethe wearer to seeabove crowds.

KEYWORD: MT7 Periscope

�1 � �2�3 � �4

1

2

34

56 8

7r

s t

7-2 Parallel and Perpendicular Lines 333

m807_c07_330_333 1/13/06 11:46 AM Page 333

Constructions

Use with Lesson 7-2

7-2

Follow the steps below to construct an angle congruent to �B.

a. Draw acute �ABC on your paper. Draw DE��.

b. With your compass point on B, draw an arc through �ABC. With the same compass opening, place your compass point on D and draw an arc through DE��. Label the intersection point F.

c. Adjust your compass to the width of the arc intersecting �ABC. Place your compass point on F and draw an arc that intersects the arc through DE�� at G. Draw DG��. Measure �ABC and �GDF.

Follow the steps below to construct parallel line segments.

1. Construct Q�R� on your paper by placing the point of your compass on Qand the pencil on R below. Draw point Q on your paper and place the point of your compass on it. Make a short arc and draw a line from Q to the arc. The intersection of the point and the arc is R. Draw point S above or below Q�R�.Draw a line through point S that intersects Q�R�. Label the intersection T.

2. Construct an angle with its vertex at S congruent to �STR. Use themethod described in . How do you know the lines are parallel?

Activity

KEYWORD: MT7 Lab7Constructing an angle is an important step in the construction ofparallel lines.

Q T R

S

Q T R

SW

Q T R

SW

CB

A

ED

CB

A

EFD

CB

A

EFD

G

2

1


1

Q R Q R

S

m807_c07_334_335 1/13/06 11:47 AM Page 334

Follow the steps below to construct perpendicular lines.

a. Draw MN�� on your paper. Draw point P b. With your compass point at P, draw an arc above or below MN��. intersecting MN�� at points Q and R.

c. Draw arcs from points Q and R, d. Draw PS��. What do you think is true about using the same compass opening, MN�� and PS�� ? Check your guess.that intersect at point S.

1. How many lines can be drawn that are perpendicular to a given line?Explain your answer.

2. Name three ways that you can determine if two lines are parallel.

Use a compass and a straightedge to construct each figure.

1. an angle congruent to �LMN 2. a line parallel to ST�� 3. a line perpendicular to GH��

4. an angle congruent to �DEF 5. a line parallel to AB�� 6. a line perpendicular to CD��

Think and Discuss

Try This

P

M N

Q

P

M N

R

Q

S

P

M N

RQ

S

P

M N

R

L

MN S

T

H

G

FE

D

B

A D

C

3


m807_c07_334_335 1/13/06 11:47 AM Page 335

Learn to find unknownangles in triangles.

Vocabulary

scalene triangle

isosceles triangle


obtuse triangle

right triangle

acute triangle

Triangle Sum Theorem

If you tear off two corners of atriangle and place them next to thethird corner, the three angles seemto form a straight angle.

Draw a triangle and extend one side.Then draw a line parallel to the extended side, as shown.

The three angles in the triangle canbe arranged to form a straight angle, or 180°.

The sides of the triangle are transversals to the parallel lines.

An has 3 acute angles. A has 1 right angle. An has 1 obtuse angle.

Finding Angles in Acute, Right, and Obtuse Triangles

Find x° in the acute triangle.

63° � 42° � x ° � 180° Triangle Sum Theorem

105° � x ° � 180°� 105° � 105° Subtract 105° from ��

x ° � 75° both sides.

Find y° in the right triangle.

37° � 90° � y° � 180° Triangle Sum Theorem

127° � y° � 180°� 127° � 127° Subtract 127°��

y° � 53° from both sides.

obtuse triangleright triangleacute triangle

TRIANGLE SUM THEOREMWords Numbers Algebra

58°

79°43° s°t°

r°The angle measuresof a triangle add to180°.

43° � 58° � 79° � 180° r° � s° � t ° � 180°

63°

42°

x°

37°y°

E X A M P L E 1


7-3 Angles inTriangles

This torn triangle demonstrates an importantgeometry theorem called the Triangle Sum Theorem.

m807_c07_336_340 1/13/06 11:53 AM Page 336

Find z° in the obtuse triangle.

13° � 62° � z ° � 180° Triangle Sum Theorem75° � z ° � 180°

� 75° � 75° Subtract 75° from��

z ° � 105° both sides.

An has 3 congruent sides and 3 congruent angles. Anhas at least 2 congruent sides and 2 congruent angles. A

has no congruent sides and no congruent angles.

Finding Angles in Equilateral, Isosceles, and Scalene Triangles

Find the angle measures in the equilateral triangle.

3m° � 180° Triangle Sum Theorem

�3m

3°

� � �18

30°� Divide both sides by 3.

m° � 60°

All three angles measure 60°.

Find the angle measures in the isosceles triangle.

55° � n° � n° � 180° Triangle Sum Theorem

55° � 2n° � 180° Simplify.� 55° � 55° Subtract 55° from both sides.��

2n° � 125°

�2

2n°� � �


n° � 62.5°

The angles labeled n° measure 62.5°.

Find the angle measures in the scalene triangle.

2p° � 3p° � 4p° � 180° Triangle Sum Theorem

9p° � 180° Simplify.

�9

9p°� � �


p° � 20°

The angle labeled 2p° measures 2(20°) � 40°, the angle labeled 3p° measures 3(20°) � 60°, and the angle labeled 4p° measures 4(20°) � 80°.

scalene triangleisosceles triangle


2E X A M P L E

13°

62°z°

m°

m° m°

55°

n° n°

4p°

3p° 2p°

7-3 Angles in Triangles 337

m807_c07_336_340 1/13/06 11:53 AM Page 337


1. Find q° in the acute triangle.

2. Find r° in the right triangle.

3. Find s° in the obtuse triangle.

4. Find the angle measures in the equilateral triangle.

5. Find the angle measures in the isosceles triangle.

6. Find the angle measures in the scalene triangle.

7. The second angle in a triangle is half as large as the first. The third angle isthree times as large as the second. Find the angle measures and draw apossible picture.

KEYWORD: MT7 Parent

KEYWORD: MT7 7-3

GUIDED PRACTICE

See Example 2

See Example 3

See Example 1

Finding Angles in a Triangle That Meets Given Conditions

The second angle in a triangle is twice as large as the first. Thethird angle is half as large as the second. Find the angle measuresand draw a possible figure.

Let x ° � first angle measure. Then 2x ° � second angle measure, and�12�(2x)° � x ° � third angle measure.

x° � 2x° � x° � 180° Triangle Sum Theorem

�44x°� � �

1840°�

Simplify, then divide bothsides by 4.

x ° � 45°

Two angles measure 45° and one angle measures 90°. The triangle hastwo congruent angles. The triangle is an isosceles right triangle.

Think and Discuss

1. Explain whether a right triangle can be equilateral. Can it beisosceles? scalene?

2. Explain whether a triangle can have 2 right angles. Can it have 2obtuse angles?

45°

15°

s°

c° c°

68°

a°

a°

a°

E X A M P L E 3

31° r°

70° 33°

q°

5d°

d°

4d°

45°

45°


m807_c07_336_340 1/13/06 11:53 AM Page 338

8. Find r ° in the acute triangle.

9. Find s ° in the right triangle.

10. Find t ° in the obtuse triangle.

11. Find the angle measures in the equilateral triangle.

12. Find the angle measures in the isosceles triangle.

13. Find the angle measures in the scalene triangle.

14. The second angle in a triangle is five times as large as the first. The third angle is two-thirds as large as the first. Find the angle measures and draw a possible picture.

Find the value of each variable.

15. 16. 17.

18. 19. 20.

Sketch a triangle to fit each description. If no triangle can be drawn, write not possible.

21. acute scalene 22. obtuse equilateral 23. right scalene

24. right equilateral 25. obtuse scalene 26. acute isosceles

27. Triangle ABC is a right triangle and m�A � 38°. What does the third angle measure?

28. Can an acute isosceles triangle have two angles that measure 40°? Explain.

29. Triangle LMN is an obtuse triangle and m�L � 25°. �M is the obtuse angle.What is the largest m�N can be to the nearest whole degree?




See Example 2

See Example 3

See Example 1

w°

w°

w°

44°

x°79°

y°

34° 34°29°

121°

w°

40°

x°

6x°

y°5y° w°

(w + 15)°45°

32°

s°r°

23° 71°

36°

m° m°

25° 40°

t°

7g°2g°

9g°

7-3 Angles in Triangles 339

m807_c07_336_340 1/13/06 11:53 AM Page 339

30. Social Studies American Samoa is aterritory of the United States made up of agroup of islands in the Pacific Ocean, abouthalfway between Hawaii and New Zealand.The flag of American Samoa is shown.

a. Find the measure of each angle in the blue triangles.

b. Use your answers to part a to find theangle measures in the white triangle.

c. Classify the triangles in the flag by theirsides and angles.

31. Choose a Strategy Which of the following sets of angle measures can be used to create an isosceles triangle?

45°, 45°, 95° 49°, 51°, 80° 27°, 27°, 126° 35°, 55°, 100°

32. Write About It Explain how to cut a square or an equilateral triangle in half to form two identical triangles. What are the angle measures in theresulting triangles in each case?

33. Challenge Find x, y, and z.

DCBA

34. Multiple Choice Which type of triangle can be constructed with a 50°angle between two 8-inch sides?

Equilateral Isosceles Scalene Obtuse

35. Short Response Two angles of a triangle are 45° and 30°. What is the measure of the third angle? Is the triangle acute, right, or obtuse?

Each square root is between two integers. Name the integers. (Lesson 4-6)

36. �42� 37. �71� 38. �35� 39. �296�

In the figure, line x⏐⏐line y. (Lesson 7-2)

40. If m�1 � 34°, what is m�7?

41. If m�6 � 125°, what is m�5?

42. If m�1 � 34°, what is m�4?

43. If m�5 � 34°, what is m�2?

DCBA

1

78 2

6 43

5

x y

d

y˚

w˚

m˚

z˚

15˚

15˚

x˚

z°x°y°

20°40°

110°110°


m807_c07_336_340 1/13/06 11:53 AM Page 340

Learn to classify andfind angles in polygons.

Vocabulary

square

rhombus

rectangle

parallelogram

trapezoid

regular polygon

polygon

E X A M P L E 1

Quadrilateral

Hexagon

Pentagon

Polygon Number of Sides

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

n-gon n

Hexagon:6 sides4 triangles

Heptagon:7 sides5 triangles

Kites have been around for over3000 years, when the Chinesemade them from bamboo and silk.The most common flat kite is inthe shape of a diamond, a type ofquadrilateral called a kite.

A is a closed plane figure formed by three or moresegments. A polygon is named by the number of its sides.

Finding Sums of the Angle Measures in Polygons

Find the sum of the angle measures in each figure.

Find the sum of the angle measures in a quadrilateral.

Divide the figure into triangles.

2 � 180° � 360° 2 triangles

Find the sum of the angle measures in a pentagon.

Divide the figure into triangles.

3 � 180° � 540° 3 triangles

Look for a pattern between the number of sides and the number of triangles.

polygon

7-4 Classifying Polygons 341


m807_c07_341_345 1/13/06 11:54 AM Page 341

The pattern is that the number of triangles is always 2 less than thenumber of sides. So an n-gon can be divided into n � 2 triangles. Thesum of the angle measures of any n-gon is 180°(n � 2).

All the sides and angles of a have equal measures.

Finding the Measure of Each Angle in a Regular Polygon

Find the angle measures in each regular polygon.

5 congruent angles 6 congruent angles

5x° � 180°(5 � 2) 6y ° � 180°(6 � 2)

5x° � 180°(3) 6y ° � 180°(4)

5x° � 540° 6y ° � 720°

�55x°� � �

5450°� �

66y°� � �

7260°�

x° � 108° y ° � 120°

Quadrilaterals with certain properties are given additional names. Ahas exactly 1 pair of parallel sides. A has 2 pairs

of parallel sides. A has 4 right angles. A has 4congruent sides. A has 4 congruent sides and 4 right angles.square

rhombusrectangleparallelogramtrapezoid

regular polygon

Quadrilaterals

Parallelograms2 pairs of parallel sides

Trapezoidsexactly 1 pair of

parallel sides

Rhombuses4 congruent sides

Rectangles4 right angles

Squares4 congruent sides

4 right angles

x°

x° x°

x° x°

y° y°

y° y°

y° y°

2E X A M P L E


m807_c07_341_345 1/13/06 11:54 AM Page 342


1. 2. 3.


4. 5. 6.

KEYWORD: MT7 Parent

KEYWORD: MT7 7-4

GUIDED PRACTICE

See Example 2

E F

H G

EF || GH

W X

Z Y

Think and Discuss

1. Choose which is larger, an angle in a regular heptagon or an anglein a regular octagon. Justify your answer.

2. Explain why all rectangles are parallelograms and why all squaresare rectangles.

Classifying Quadrilaterals

Give all of the names that apply to each figure.

quadrilateral Four-sided polygon

trapezoid 1 pair of parallel sides

quadrilateral Four-sided polygon

parallelogram 2 pairs of parallel sides

rectangle 4 right angles

E X A M P L E 3

t° t°

t° t°

v° v°

v°v°

v°v° v°

b° b°

b° b°

b° b°

See Example 1



m807_c07_341_345 1/13/06 11:54 AM Page 343


7. 8. 9.


10. 11. 12.


13. 14. 15.


16. 17. 18.

Find the sum of the angle measures in each regular polygon. Then find themeasure of each angle.

19. 20-gon 20. 13-gon 21. 60-gon 22. pentagon 23. 16-gon


24. 25. 26.

27. 28. 29.




See Example 2

See Example 3

See Example 1

See Example 3

4 cm4 cm

4 cm4 cm

m° m°

m° m°

m° m°

m° m°

h°h°h°

h°h°

h° h° h°h°h°

h°h° v°

v°

v°

v°

v°

v°

v°

v°v° v°

8 in.8 in.

8 in.8 in.

A C

D

B

AB || CDAD || BC

P Q

R S

PQ || RS

x°

110° 45°

75°y°

60°

25°w°

55°

113°

110°125°

z°

123°

133°

150°

116°101°

x° x°

40° 40°m°

4m°50°

110°


m807_c07_341_345 1/13/06 11:54 AM Page 344

The sum of the angle measures of a polygon is given. Name the polygon.

30. 1080° 31. 540° 32. 360° 33. 1620°

Graph the given vertices on a coordinate plane. Connect the points to draw apolygon and classify it by the number of its sides.

34. A(0, 3), B(1, 1), C(4, 1), D(5, 3), E(4, 5), F(1, 5)

35. A(�3, 2), B(�3, �1), C(1, �3), D(4, 0), E(1, 4)

Sketch a quadrilateral to fit each description. If no quadrilateral can be drawn, write not possible.

36. a parallelogram that is not a rhombus

37. a square that is not a rectangle

38. Earth Science Precious stones are oftencut in a brilliant cut to maximize the lightthey reflect. The best angles for a cutdepend on the type of stone. The bestangles for a diamond are shown.

a. If the pavilion main angle is 41°, find x.

b. If the crown angle is 35°, find y.

39. Architecture Fernando is designing a house. He wants one room to be in theshape of an irregular heptagon with two corners that form right angles. Whatangle measures could the remaining five corners have?

40. What’s the Error? A student said that all squares are rectangles, but not all squares are rhombuses. What was the error?

41. Write About It Why is it possible to find the sum of the angle measures of an n-gon using the formula (180n � 360)°?

42. Challenge Use a diagram and the properties of parallel lines to explainwhich angles in a parallelogram must be congruent.

43. Multiple Choice What is the measure of each angle of a regular 15-sided polygon?

146° 148° 150° 156°

44. Short Response The sum of the angle measures of a regular polygon is 720°. Name the regular polygon. What is the measure of each angle?

Subtract. (Lesson 1-5)

45. �5 � 12 46. �25 � 25 47. 34 � (�17) 48. �30 � 41

49. The first angle in a triangle is less than 90°. The second angle is �34� as large as

the first angle. The third angle is �23� as large as the second angle. Find the angle

measures and draw a possible figure. (Lesson 7-3)

DCBA

y ˚y ˚

x ˚

35˚

41˚ 41˚

35˚Crown angle

Pavilion mainangle

Earth Science

The ImperialState Crown ofGreat Britaincontains over3000 preciousstones, including2800 diamonds.


m807_c07_341_345 1/13/06 11:54 AM Page 345

Exterior Angles of a Polygon

Use with Lesson 7-4

7-4

The exterior angles of a polygon are formed by extending the polygon’s sides. Every exterior angle is supplementary to the angle next to it inside the polygon.

Follow the steps to find the sum of the exterior angle measures for a polygon.

a. Use geometry software to make a pentagon. b. Use the LINE-RAY tool to extend Label the vertices A through E. the sides of the pentagon. Add

points F through J as shown.

c. Use the ANGLE MEASURE tool to measure d. Drag vertices A through E and watch the each exterior angle and the CALCULATOR sum. Notice that the sum of the angle tool to add the measures. Notice the sum. measures is always 360°.

1. Suppose you were to drag the vertices of a polygon so that the polygon almost vanishes. How would this show that the sum of the exterior angle measures is 360°.

1. Use geometry software to draw any polygon. Find the sum of its exteriorangle measures. Drag its vertices to check that the sum is always the same.

Activity

KEYWORD: MT7 Lab7

Exterior angle

Think and Discuss

Try This

1


m807_c07_346 1/13/06 11:54 AM Page 346

Learn to identifypolygons in thecoordinate plane.

Vocabulary

run

rise

slope

C

D

–3

6

Negative slope

x

y

L

MK N

x

y

In computer graphics, a coordinate system is used to create images, from simple geometric figures to realistic figures used in movies.

Properties of the coordinate plane can be used to find information about figures in the plane, such as whether lines in the plane are parallel.

is a number that describes how steep a line is.

slope � � �r

r

i

u

s

n

e�

slope of A�B� � �86� � �

43�

slope of C�D� � ��63� � �

�21�

The slope of a horizontal line is 0. The slope of a vertical line is undefined.

Finding the Slope of a Line

Determine if the slope of each line is positive, negative, 0, or undefined. Then find the slope of each line.

KL��

positive; slope of KL�� 21� � 2

LM��

undefined; slope of LM�� 10�

LN��

negative; slope of LN�� 42� � ��

12�

KM��

0; slope of KM�� 01�

vertical change��horizontal change

Slope

When a number is divided by zero,the quotient isundefined. There is no answer.

7-5 Coordinate Geometry 347


A

B

Positive slope

x

y

E X A M P L E 1

m807_c07_347_351 1/19/06 4:07 PM Page 347

Finding Perpendicular and Parallel Lines

Which lines are parallel? Which lines are perpendicular?

slope of PQ�� 32�

slope of RS�� 43�

slope of AB�� 32�

slope of PA�� 22� or �1

slope of GH�� 43�

slope of XY�� 87�

PQ��⏐⏐AB�� The slopes are equal: �32� � �

32�

RS�� ⊥ GH�� The slopes have a product of �1: �43� � �

�43� � �1

Using Coordinates to Classify Quadrilaterals

Graph the quadrilaterals with the given vertices. Give all of thenames that apply to each quadrilateral.

J(1, 2), K(4, 2), P(�1, 2), Q(2, 1),L(4, �1), M(1, �1) R(�1, �2), S(�3, 0)

J�K�⏐⏐M�L� and M�J�⏐⏐L�K� S�P�⏐⏐R�Q�J�K� ⊥ L�K�, J�K� ⊥ M�J�, trapezoid

M�L� ⊥ L�K� and M�L� ⊥ M�J�parallelogram, rectangle, square, rhombus

A

x

y

X

S

Q

BP

H

G

Y

R

y

Undefinedslope

Undefinedslope

Slope = 0

Slope = 0M

KJ

L

x x

y

P

S

R

Slope = – 13

Slope = 1

Slope = 1

Slope = –1

Q

If a line has slope

�ba

�, then a line

perpendicular to it

has slope ��ba�.

Slopes of Parallel and Perpendicular Lines

Two lines with equal slopes are parallel.

Two lines whose slopes have a product of �1 are perpendicular.

2E X A M P L E

E X A M P L E 3


m807_c07_347_351 1/13/06 11:55 AM Page 348


Determine if the slope of each line is positive,negative, 0, or undefined. Then find the slope ofeach line.

1. AD�� 2. BE��

3. MN�� 4. EF��

5. Which lines are parallel?

6. Which lines are perpendicular?

Graph the quadrilaterals with the given vertices. Giveall of the names that apply to each quadrilateral.

7. D(�3, �2), E(�3, 3), F(2, 3), G(2, �2)

8. R(�4, �1), S(�2, 2), T(4, 2), V(5, �1)

Find the coordinates of the missing vertex.

9. rhombus ABCD with A(2, 3), B(3, 1), and D(1, 1)

10. square JKLM with J(�3, 1), K(0, 1), and L(0, �2)

KEYWORD: MT7 Parent

KEYWORD: MT7 7-5

GUIDED PRACTICE

See Example 2

See Example 3

See Example 1

Finding the Coordinates of a Missing Vertex

Find the coordinates of the missing vertex of square ABCD.Square ABCD with A(4, 0), B(0, 4),and C(�4, 0)

Step 1 Graph and connect thegiven points.

Step 2 Complete the figure to find the missing vertex.A�B� has a slope of �1, so C�D� has a slope of �1. B�C�has a slope of 1, so A�D�has a slope of 1.

The coordinates of D are (0, �4).

C A

D

B

O 4

4

2

2�2

�2

�4

�4

x

y

Think and Discuss

1. Explain how you can use slopes to classify a quadrilateral.

4E X A M P L E

x

y

D

M

N

A

F B

E

C

In a square oppositesides are parallel.


See Example 4

m807_c07_347_351 1/13/06 11:55 AM Page 349

Determine if the slope of each line is positive,negative, 0, or undefined. Then find the slope of each line.

11. AB�� 12. EG��

13. HG�� 14. CH��

15. Which lines are parallel?

16. Which lines are perpendicular?

Graph the quadrilaterals with the given vertices.Give all of the names that apply to each quadrilateral.

17. D(�4, 3), E(4, 3), F(4, �5), G(�4, �5)

18. W(�2, 1), X(�2, �2), Y(4, 1), Z(0, 2)


19. rectangle ABCD with A(�3, 3), B(4, 3), and D(�3, �1)

20. trapezoid JKLM with J(�1, 5), K(2, 3), and L(2, 1)

Draw the line through the given points and find its slope.

21. A(1, 0), B(2, 3) 22. C(�3, 0), D(�3, �4)

23. G(4, �2), H(�1, �2) 24. E(�2, 1), F(3, �2)

25. A line passes through the coordinates P(1, 3) and Q(�2, �3). Identify the slope of PQ��. Then name two coordinates and the slope of a line perpendicular to PQ��.

26. AB��⏐⏐CD�� and the slope of AB�� is undefined. What can you tell about the slope of CD��? Explain.

27. On a coordinate grid draw a line s with slope 0 and a line t with slope 1. Then draw three lines through the intersection of lines s and t that have slopes between 0 and 1.

28. On a coordinate grid draw a line m with slope 0 and a line n with slope �1.Then draw three lines through the intersection of lines m and n that haveslopes between 0 and �1.

29. Critical Thinking Square ABCD has vertices at (1, 2) and (1, �2). Findthe possible coordinates of the two missing vertices to create the squarewith the least area. Justify your solution.

30. Critical Thinking Triangle LMN has vertices at L(�2, 2), M(0, 0), andN(�5, �1). What kind of triangle is it? Explain.




See Example 2

See Example 3

See Example 4

See Example 1

x

y

H

D

C

B

E

A

GF


m807_c07_347_351 1/19/06 4:07 PM Page 350

Tell if each statement is true or false. If it is false, give a counterexample.

31. Opposite sides of a rhombus have the same slope.

32. All of the adjacent sides of quadrilaterals have slopes with a product of �1.

33. All parallelograms have two pairs of lines with the same slope and adjacent sides that have slopes with a product of �1.

34. A trapezoid has two pairs of sides that have the same slope.

35. The slope of a horizontal line is always 0.

36. The slope of a line through the origin is always defined.

Identify and name each figure.

37. This figure has two sides with undefined slopes.

38. This figure has a side with a slope of �1.

39. This figure has a side with a slope of 3.

40. This figure has a side with a slope of �13�.

41. What’s the Question? Points P(3, 7),Q(5, 2), R(3, �3), and S(1, 2) form thevertices of a polygon. The answer is thatthe segments are not perpendicular. What is the question?

42. Write About It Explain how using different points on a line to find theslope affects the answer.

43. Challenge Use a square in a coordinate plane to explain why a line withslope 1 makes a 45° angle with the x-axis.

44. Multiple Choice A right triangle has vertices at (0, 0), (0,4), and (10, 4). What is the slope of the hypotenuse?

2.5 2 1.8 0.4

45. Gridded Response Find the slope of the line that crosses through the points A(2, 4) and B(�1, 5).

Find each number. (Lesson 6-4)

46. 60% of what number is 12? 47. 112 is 80% of what number?

48. 30 is 2% of what number? 49. 90% of what number is 18?

Find the sum of the angle measures of each polygon. (Lesson 7-4)

50. 15-gon 51. hexagon 52. n-gon 53. decagon

DCBA

x

y

4 5321

345

21

�3�4�5 �2�1

�2�1

�3�4�5

O

A

B

D

C


m807_c07_347_351 1/13/06 11:55 AM Page 351

Quiz for Lessons 7-1 Through 7-5

7-1 Points, Lines, Planes, and Angles


1. two pairs of complementary angles

2. three pairs of supplementary angles

3. two right angles

7-2 Parallel and Perpendicular Lines

In the figure, line m⏐⏐line n. Find the measure of each angle.

4. �1 5. �2 6. �3

7-3 Angles in Triangles

Find x° in each triangle.

7. 8.

9. In �ABC, m�A � 57°, and �B is a right angle. What is m�C?



10. 11.


Graph the quadrilaterals with the given vertices. Give all of the names that apply to each quadrilateral.

12. A(�2, 1), B(3, 2), C(2, 0), D(�3, �1) 13. P(�3, 4), Q(2, 4), R(2, �1), S(�3, �1)


14. square ABCD with A(�1, 1), B(2, 1), and C(2, �2)

15. parallelogram PQRS with P(3, 3), Q(4, 2), and R(2, �2)

Rea

dy

to G

o O

n?


CB

D

A

E

F

35°

15°

75° 55°

m

n

125°

3 4

1 2

83°

x°60°

74°

x° x°

3 in.3 in.

3 in.

P

N

OM

3 in.A B

D C

AB || CD

m807_c07_352 1/13/06 11:55 AM Page 352

In the figure, �1 and �2 are complementary, and �1 and �5 are supplementary. If m�1 � 60°, find m�3 � m�4.

In triangle ABC, m�A � 35° and m�B � 55°. Use the Triangle Sum Theorem to determine whether triangle ABC is a right triangle.

The second angle in a quadrilateral is eighttimes as large as the first angle. The thirdangle is half as large as the second. Thefourth angle is as large as the first angle andthe second angle combined. Find the anglemeasures in the quadrilateral.

Parallel lines m and n are intersected by atransversal, line p. The acute angles formed by line m and line p measure 45°.Find the measure of the obtuse anglesformed by the intersection of line n and line p.

Write each problem in your own words. Check to make sure youhave included all of the information needed to solve the problem.

Understand the Problem• Restate the problem in your own words

If you write a problem in your own words, you may understand itbetter. Before writing a problem in your own words, you may needto read it over several times—perhaps aloud, so you can hearyourself say the words.

Once you have written the problem in your own words, you maywant to make sure you included all of the necessary information tosolve the problem.

Focus on Problem Solving 353

m

n

p

34

51 2

55°35°A

C

B

2

3

4

1

m807_c07_353 1/13/06 11:55 AM Page 353

Learn to use propertiesof congruent figures tosolve problems.

Vocabulary

correspondence

Below are the DNA profilesof two pairs of twins. TwinsA and B are identical twins.Twins C and D are fraternaltwins.

A is a way of matching up two sets of objects. The bands of DNA that are next to each other in each pair match up, or correspond. In the DNA of the identical twins, the corresponding bands are the same.

If two polygons are congruent, all of their corresponding sides and angles are congruent.

Writing Congruence Statements

Write a congruence statement for each pair of congruent polygons.

In a congruence statement, the vertices in the second triangle haveto be written in order of correspondence with the first triangle.

�K corresponds to �R. �K � �R�L corresponds to �Q. �L � �Q�M corresponds to �S. �M � �S

The congruence statement is triangle KLM � triangle RQS.

correspondence

A

B

C

D

65°

94°

21°

K

M L

65°

94°21°

Q S

R

E X A M P L E 1

Marks on the sides ofa figure can be usedto show congruence.K�M� � R�S� (1 mark)K�L� � R�Q� (2 marks)M�L� � S�Q� (3 marks)


7-6 Congruence

CONGRUENT TRIANGLES

Corresponding CorrespondingDiagram Statement Angles Sides

�ABC � �DEF �A � �D A�B� � D�E�

�B � �E B�C� � E�F�

�C � �F A�C� � D�F�

A B

C

D

EF

m807_c07_354_357 1/13/06 11:55 AM 4


The vertices in the first pentagon are written in order around thepentagon starting at any vertex.

�A corresponds to �H. �A � �H�B corresponds to �I. �B � �I�C corresponds to �J. �C � � J�D corresponds to �F. �D � �F�E corresponds to �G. �E � �G

The congruence statement is pentagon ABCDE � pentagon HIJFG.

Using Congruence Relationships to Find Unknown Values

In the figure, quadrilateral PQSR � quadrilateral WTUV.

Find x. Find y.x � 5 � 12 P�R� � W�V� 6y � 24 W�T� � P�Q�

�5 � �5�66y� � �

264��

x � 7y � 4

Find z.132 � 11z �R � �V

�11312

� � �1111z

� Divide both sides by 11.

12 � z

Think and Discuss

1. Explain what it means for two polygons to be congruent.

2. Tell how to write a congruence statement for two polygons.

A

CB

DE130° 100°

95°

80°135°G

H

I

FJ

130°

100°80°

135°

95°

2E X A M P L E

P Q

R

70°

132°

75°

x � 518

24

S

W

6y

12

T

VU(11z)°

75°

Divide bothsides by 6.

Subtract 5 fromboth sides.

7-6 Congruence 355

m807_c07_354_357 1/13/06 11:55 AM Page 355



1. 2.

In the figure, triangle ABC � triangle LMN.

3. Find q. 4. Find r. 5. Find s.


6. 7.

In the figure, quadrilateral ABCD � quadrilateral LMNO.

8. Find m. 9. Find n. 10. Find p.


11. pentagon ABCDE � 12. hexagon ABCDEF �pentagon PQRST hexagon LMNOPQ

KEYWORD: MT7 Parent

KEYWORD: MT7 7-6

GUIDED PRACTICE




See Example 2

See Example 2

See Example 1

See Example 1

B C

A 58°

32°

C

A

65°

28

r � 9

B

L

N M25°

13q°

4s

13

E

F

D 32°

58°

E F

D

42°

123° G

HI

123°

15°

N

M

O

L

92°144°

75°49°

R

S

Q

T

92° 144°

75°49°

DC

E

Q

154°

116°

118°

(180 � pA F

B

D

E

Q

8°

(180 � p)°

(n � 86)°

(11m)°

F ML

N

OP

P

Q

S R

75°Y X

Z W

105°

CD

131° 11

74°

3p

AB

O

N

L M(m � 75)°

n + 421

27

AB

E D

C

(80 � y)°

6x°6.5

13.6

5.7Q

S

P

R

T114°

107°

z – 4.5


m807_c07_354_357 1/13/06 11:55 AM Page 356


13. quadrilateral ABCD � 14. heptagon ABCDEFG �quadrilateral EFGH heptagon JKLMNOP

15. Right triangle PQR � right triangle STU. m�P � 28° and m�U � 90°. Find m�Q.

16. Write a Problem Write and solve a problem about the right triangles shown.

17. Write About It How can knowing two polygons are congruent help youfind angle measures of the polygons?

18. Challenge Triangle ABC � triangle LMNand A�E�⏐⏐B�D�. Find m�ACD.

19. Multiple Choice Triangle EFG �triangle JIH. Find the value of x.

5.67 30 63 71

20. Multiple Choice Triangle ABC � triangle JKL. m�A � 30° and m�B � 50°. Find the m�K.

30° 50° 80° 100°

21. Gridded Response Quadrilateral ABCD � quadrilateral WXYZ. The length of A�B� � 21 and the length of W�X� � 7m. Find m.

Find the missing y-coordinate of each ordered pair that is a solution to y � 4x � 2. (Lesson 3-2)

22. (0, y) 23. (1, y) 24. (3, y) 25. (7, y)

The measures of two angles of a triangle are given. Find the measure of thethird angle. (Lesson 7-3)

26. 45°, 45° 27. 30°, 60° 28. 21°, 82° 29. 105°, 42°

JHGF

DCBA

14.2y

21.3

18.9

G

F

E

9w 7.6z

0.3xI

H

J

BC

A

D

F

G

E

(w � 88)°

102°123°

127°

AB

D

C65

3r

78

G

H

FE

s � 55

t � 30

72

56 139°

128°

N

O

M

L

J

K

P °x4

y3 °

65°

25°C

B

A

65°

25°F

E

D

BC

E

D

A L

N M

20°

80°

7-6 Congruence 357

m807_c07_354_357 1/19/06 4:07 PM Page 357

Learn to transformplane figures usingtranslations, rotations,and reflections.

Vocabulary

image

reflection

center of rotation

rotation

translation

transformation

When you are on an amusementpark ride, you are undergoing atransformation. A is a change in a figure’s position orsize. Ferris wheels and merry-go-rounds are rotations. Free-fall ridesand water slides are translations.Translations, rotations, andreflections are types oftransformations.

The resulting figure, or , of a translation, rotation, or reflectionis congruent to the original figure.

Identifying Transformations

Identify each as a translation, rotation, reflection, or none of these.

translation none of these

\

rotation reflection

image

transformation

Translation Rotation Reflection

A slides a A turns a A flips a figure along a line figure around a point, figure across a line to without turning. called the create a mirror image.

.rotationcenter of

reflectionrotationtranslation

A� is read “A prime.”The point A� is theimage of point A.

E X A M P L E 1

P N

MKL

A BCF

DE

A′ B′C′

D′E′F′

J

H

G

E

FG′

H′

J′E′

F′

Q T

RS

P′

L′

K′

M′ N′

T′

S′R′

Q′


7-7 Transformations

m807_c07_358_361 1/13/06 11:56 AM Page 358

Graphing Transformations

Draw the image of a triangle with vertices A(1, 1), B(1, 4), and C(3, 4) after each transformation.

translation 5 units down reflection across the y-axis

Describing Graphs of Transformations

Parallelogram EFGH has vertices E(�2, 1), F(3, 1), G(4, 4), and H(�1, 4). Find the coordinates of the image of the indicated pointafter each transformation.

translation 2 units down, 180° rotation around (0, 0),point E point G

E� (�2, �1) G� (�4, �4)

Think and Discuss

1. Tell whether the image of a vertical line is sometimes, always, ornever vertical after a translation, a reflection, or a rotation.

2. Describe what happens to the x-coordinate and the y-coordinateafter a point is reflected across the x-axis.

2E X A M P L E

E X A M P L E 3

x

y

�2�4

�2

�4

2

4

2

A

B C

A′

B′ C′

x

y

�2�4

�2

�4

2

4

2 4

A

B C

A′

C′ B′

H G

E

O 42

�2

�4

�4

x

y

FH′

E′

G′

F′

H G

E

O

2

2�2

�2

�4

�4

x

y

F

E′F′

G′ H′

7-7 Transformations 359

m807_c07_358_361 1/13/06 11:56 AM Page 359



1. 2.

Draw the image of the parallelogram ABCD with vertices (�3, 0), (�4, 3), (1, 4),and (2, 1) after each transformation.

3. translation 1 unit up 4. reflection across the x-axis

5. reflection across the y-axis 6. 180° rotation around (0, 0)

Triangle ABC has vertices A(2, 1), B(3, 3), and C(1, 2). Find the coordinatesof the image of the indicated point after each transformation.

7. translation 4 units down, point C 8. reflection across the x-axis, point B

9. reflection across the y-axis, point C 10. 180° rotation around (0,0), point A


11. 12.

Draw the image of the quadrilateral ABCD with vertices (1, 1), (2, 4), (4, 5),and (5, 3) after each transformation.

13. translation 5 units down 14. reflection across the x-axis

15. reflection across the y-axis 16. 180° rotation around (0, 0)

Square ABCD has vertices A(�2, 2), B(2, 2), C(2, �2), and D(�2, �2). Find thecoordinates of the image of the indicated point after each transformation.

17. translation 3 units to the left, point A

18. translation 4 units to the right, point B

19. reflection across the x-axis, point C

20. 180° rotation around (0, 0), point A

KEYWORD: MT7 Parent

KEYWORD: MT7 7-7

GUIDED PRACTICE


See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

Q

U T

R

S

T′ U′

S′

R′Q′

J

L

M

K

M′K′

J′

L′

BC

A

C′

A′

B′

ZX

Y

X′

Y′

Z′


m807_c07_358_361 1/13/06 11:56 AM Page 360

36. Multiple Choice Which best represents the transformation at right?

Translation

Rotation

Reflection

None of these

37. Short Response Draw the image of a triangle with vertices (�1, 2), (3, 3), and (1, �3) after a translation 2 units up and 2 units to the right.

Find each percent increase or decrease to the nearest percent. (Lesson 6-5)

38. from 75 to 90 39. from 1200 to 1400 40. from 44 to 21

Draw the line through the given points and find its slope. (Lesson 7-5)

41. A(5, 2), B(3, 2) 42. G(6, �3), H(�4, �9) 43. C(3, 4), D(0, 0)

D

C

B

A

Copy each figure and perform the given transformations.

21. Reflect across line m. 22. Reflect across line n 23. Rotate clockwise 90°.

Give the coordinates of each point after a reflection across the given axis.

24. (1, 4); x-axis 25. (�3, 2); x-axis 26. (m, n); x-axis

27. (5, �2); y-axis 28. (�2, 4); y-axis 29. (m, n); y-axis

Give the coordinates of each point after a 180° rotation around (0, 0).

30. (1, 2) 31. (�4, 5) 32. (m, n)

33. Write a Problem Write a problem involving transformations on acoordinate grid that result in a pattern.

34. Write About It Explain how each type oftransformation performed on the arrow would affect the direction the arrow is pointing.

35. Challenge A triangle has vertices (2, 5), (3, 7), and (7, 5). After a reflectionand a translation, the coordinates of the image are (7, �2), (8, �4), and(12, �2). Describe the transformations.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page 795.

m

n

Fm

n

L m

n

D

O 4

4

2�2

x

y

7-7 Transformations 361

m807_c07_358_361 1/13/06 11:56 AM Page 361

Combine Transformations

Use with Lesson 7-7

7-7

KEY

Pattern blocks �

triangle rhombus trapezoid

You can use a coordinate plane when transforming a geometric figure.

Follow the steps below to transform a figure.

a. Place a red pattern block on a coordinateplane. Trace the block, and label the vertices.

b. Translate the figure 3 units down and 5 unitsright, and then reflect the resulting figureacross the x-axis. Draw the image and labelthe vertices.

c. Now place a green pattern block on the samecoordinate plane. Trace the block and labelthe vertices. Rotate the figure 180° around thepoint (0, 0), and then translate it 4 units upand 3 units right. Draw the image and labelthe vertices.

1. When you perform two or more transformations on a figure, does itmatter in which order the transformations are performed? Explain.

1. Place a blue pattern block on a coordinate plane. Trace the block, andlabel the vertices. Perform two different transformations on the figure.Draw the image and label the vertices. Trade with a classmate.Describe the transformations your classmate used.

Activity 1

KEYWORD: MT7 Lab7

1

O 4 6 8

4

6

8

2

2�2

�2

�4

�6

�8

x

y

10 12

Think and Discuss

Try This


m807_c07_362_363 1/13/06 11:56 AM Page 362

Follow the steps below to transform a figure.

a. Place a rhombus on a coordinate plane. Trace the rhombus, and label the vertices.

b. Rotate the figure 90° clockwise about the origin.

c. Reflect the resulting figure across the x-axis. Draw the image and label the vertices.

d. Now place a rhombus in the same position as the original figure. Reflect the figure across the line y � x.

1. What do you notice about the images that result from the twotransformations in parts b and c above and the image that resultsfrom the single transformation in part d above?

1. Place a pattern block on a coordinate plane. Trace the block and labelthe vertices. Perform two different transformations on the figure. Drawthe image and label the vertices. Explain what single transformation ofthe original figure would result in the same image.

Describe two different ways to transform each figure from position A to position B.

2. 3.

O 4

2

�2

�4

�4

x

A

y

BO 4

4

2

2�2

�2

�4

�4

x

A

y

B

Activity 2

1

Think and Discuss

Try This

O 4

4

2

2�2

�2

�4

�4

x

y

O 4

4

2

2�2

�2

�4

�4

x

y


m807_c07_362_363 1/13/06 11:56 AM Page 363

Learn to identifysymmetry in figures.

Vocabulary

rotational symmetry

line of symmetry

line symmetry

E X A M P L E 1

If you fold a figureon the line ofsymmetry, the halvesmatch exactly.

Nature provides many beautiful examples ofsymmetry, such as the wings of a butterfly orthe petals of a flower. Symmetric objects haveparts that are congruent.

A figure has if you can draw aline through it so that the two sides are mirrorimages of each other. The line is called the

.

Drawing Figures with Line Symmetry

Complete each figure. The dashed line is the line of symmetry.

line of symmetry

line symmetry

7-8 Symmetry


m807_c07_364_367 1/13/06 11:56 AM Page 364

A figure has if you can rotate the figure aroundsome point so that it coincides with itself. The point is the center ofrotation, and the amount of rotation must be less than one full turn,or 360°.

7-fold rotational symmetry 6-fold rotational symmetry

Drawing Figures with Rotational Symmetry

Complete each figure. The point is the center of rotation.

2-fold

Figure coincides with itself twice every full turn.

8-fold

Figure coincides with itself 8 times every full turn.

rotational symmetry

Think and Discuss

1. Explain what it means for a figure to be symmetric.

2. Tell which letters of the alphabet have line symmetry.

3. Tell which letters of the alphabet have rotational symmetry.

2E X A M P L E

7-fold and 6-foldrotational symmetrymean that the figurescoincide with themselves7 times and 6 timesrespectively, within onefull turn.

7-8 Symmetry 365

m807_c07_364_367 1/19/06 4:08 PM Page 365



1. 2. 3. 4.


5. 4-fold 6. 6-fold 7. 3-fold


8. 9. 10.

11. 12. 13.


14. 4-fold 15. 5-fold 16. 2-fold

KEYWORD: MT7 Parent

KEYWORD: MT7 7-8

GUIDED PRACTICE


See Example 2

See Example 1

See Example 1

See Example 2


m807_c07_364_367 1/13/06 11:56 AM Page 366

Kage Asa no ha Maru ni shichiyo Nito Nami

27. Short Answer Draw a figure that has line symmetry and rotational symmetry.

28. Multiple Choice Which figure has 90° rotational symmetry?

regular pentagon regular hexagon

square regular heptagon

Find each unit rate. (Lesson 5-2)

29. 20 bananas for $4.40 30. 496 miles in 16 hours 31. 20 oz for $3.20

In the figure, ABCDE � PQRST. (Lesson 7-6)

32. Find j. 33. Find k.

34. Find m. 35. Find n.

DB

CA

Draw an example of a figure with each type of symmetry.

17. line and rotational symmetry 18. no symmetry

How many lines of symmetry do the following figures have?

19. square 20. rectangle

21. equilateral triangle 22. isosceles triangle

23. Social Studies Family crests called ka-mon have been in use in Japan formany centuries. Copy each crest below. Describe the symmetry, and drawany lines of symmetry or the center of rotation.

a. b. c.

24. Write a Problem Signal flags are hung from lines of rigging on ships.Write a problem about the types of symmetry in the flags.

25. Write About It To complete a figure with n-fold rotational symmetry,explain how much you rotate each part.

26. Challenge The flag of Switzerland has 180°rotational symmetry. Identify at least three other countries that have flags with 180°rotational symmetry.


See page 795.

EB

D

A

C

4m � 3

2.5k

21

2j � 4T

Q

S

P

R

13

10

3n

14

7-8 Symmetry 367

Social Studies

In Japan, akimono thatdisplays thewearer’s familycrest is wornfor ceremonialoccasions.

m807_c07_364_367 1/13/06 11:56 AM Page 367

Learn to createtessellations.

Vocabulary

regular tessellation

tessellation

Fascinating designs can be made byrepeating a figure or group of figures.These designs are often used in art andarchitecture.

A repeating pattern of plane figures thatcompletely covers a plane with no gapsor overlaps is a .

In a , a regularpolygon is repeated to fill a plane. Theangle measures at each vertex must add to 360°, so only three regulartessellations exist.

Equilateral triangles Squares Regular hexagons

6 � 60° � 360° 4 � 90° � 360° 3 � 120° � 360°

It is also possible to tessellate with polygons that are not regular. Sincethe angle measures of a triangle add to 180°, six triangles meeting ateach vertex will tessellate. The angle measures of a quadrilateral add to360°, so four quadrilaterals meeting at a vertex will tessellate.

Creating a Tessellation

Create a tessellation withquadrilateral ABCD.

There must be a copy of each angleof quadrilateral ABCD at every vertex.


tessellation

E X A M P L E 1B

A

D C


7-9 Tessellations

Alcazar Palace in Seville, Spain

m807_c07_368_371 1/13/06 11:57 AM Page 368

Think and Discuss

1. Explain why a regular pentagon cannot be used to create a regulartessellation.

2. Describe the transformations used to make the tessellation inExample 2.

2E X A M P L E

1. Create a tessellation with quadrilateral QRST.

2. Use rotations to create a variation of the tessellation in Exercise 1.

KEYWORD: MT7 Parent

KEYWORD: MT7 7-9

GUIDED PRACTICE

See Example 2

See Example 1

Q

S

T

R

Creating a Tessellation by Transforming a Polygon

Use rotations to create a variation of the tessellation in Example 1.Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.

Step 4: Repeat with the other sides.

Step 5: Use the figure to make a tessellation.

BA

D C

BA

D C

BA

D C

7-9 Tessellations 369


m807_c07_368_371 1/13/06 11:57 AM Page 369

Use each shape to create a tessellation.

5. 6. 7.

8. 9. 10.

11. A piece is removed from one side of a rectangle and translated to the opposite side. Will this shape tessellate?

12. A piece is removed from one side of a trapezoid and translated to the opposite side. Will this shape tessellate?

13. In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical. Use each arrangement of regular polygons to create a semiregular tessellation.

a. b. c.


See page 795.

3. Create a tessellation with triangle PQR.

4. Use rotations to create a variation of the tessellation in Exercise 3.


See Example 2

See Example 1 Q

RP


m807_c07_368_371 1/13/06 11:57 AM Page 370

Art

Hand

with

refle

ctin

g sp

here

by

M.C

.Esc

her ©

2004

Cord

on A

rt-B

aarn

-Hol

land

.All

right

s re

serv

ed.

M. C. Escher created works of art by repeating interlocking shapes.He used both regular and nonregular tessellations. He often usedwhat he called metamorphoses, in which shapes change into othershapes. Escher used his reptile pattern in many hexagonaltessellations. One of the most famous is entitled simply Reptiles.

14. The steps below show the method Escher used to make abird out of a triangle. Use the bird to create a tessellation.

15. Critical Thinking What regular polygon do you think Escher used to begin Reptiles?

16. Challenge Create an Escher-like tessellation of your own design.

Sym

met

ry D

raw

ing

E25

by M

.C.E

sche

r ©20

04

Cord

on A

rt-B

aarn

-Hol

land

.All

right

s re

serv

ed.

Step 1 Step 2

Step 3 Step 4

7-9 Tessellations 371

KEYWORD: MT7 Escher

17. Multiple Choice Which of the following shapes will NOT form a regular tessellation?

18. Short Answer Which set of polygons will create a tessellation? Explain.

Write each number in scientific notation. (Lesson 4-4)

19. 3,400,000,000 20. 0.00000045 21. 28,000

Tell whether the two lines described in each exercise are parallel, perpendicular, orneither. (Lesson 7-5)

22. PQ�� has slope �32�. EF�� has slope ��

23�. 23. AB�� has slope �1

91�. CD�� has slope ��

34�.

DCBA

m807_c07_368_371 1/13/06 11:57 AM Page 371

Quiz for Lessons 7-6 Through 7-9

7-6 Congruence

In the figure, triangle ABC � triangle LMN.

1. Find q. 2. Find r. 3. Find s.

7-7 Transformations


4. 5.

Quadrilateral ABCD has vertices A(�7, 5), B(�4, 5), C(�2, 3), and D(�6, 2).Find the coordinates of the image of each point after each transformation.

6. translation 4 units down, point C 7. reflection across the y-axis, point A

7-8 Symmetry

8. Complete the figure. The dashed line 9. Complete the figure with 4-fold is the line of symmetry. rotational symmetry. The point is

the center of rotation.

7-9 Tessellations

10. Copy the given figure and use it to create a tessellation.

Rea

dy

to G

o O

n?


C N M

A

65°

25°

135q°

28 7s

r � 9

B

L

U

T

RQ

S

U′T′

Q′R′

S′

KN

L

J

M

N′

L′M′

J′

K′

m807_c07_372_372 1/13/06 11:57 AM Page 372

Mu

lti-Step Test Prep

Multi-Step Test Prep 373

Cloth Creations The Asante people of Ghana are knownfor weaving Kente cloth, a colorful textile based on repeatinggeometric patterns. Susan is using a coordinate plane todesign her own Kente cloth pattern.

1. Susan starts with triangle ABCas shown. Explain how she canuse slopes to make sure thetriangle is a right triangle.

2. To begin the pattern, Susan uses transformations to make a row of triangles that are allcongruent to triangle ABC.Describe the transformations she should use to make these triangles.

3. Next, she extends the pattern by making additionalrows of triangles. The firsttriangle in Row 2 is shown.Complete the table by writingthe coordinates of the topvertex of each triangle in the pattern.

4. What patterns do you notice in the table?

5. Susan’s Kente cloth pattern is a tessellation of what types of figures?

x

y

22242628210

2

2 4 6O

C

A B

x

y

22242628210

2

2 4 6O

C

A B

x

y

22242628210

4

6

2

2 4 6O

Row 4

Row 3

Row 2

Row 1

C

A B

Row Top Vertex of Triangles in Row

Row 1 (�9, 2) (�4, 2) (1, 2) (6, 2)

Row 2 (�8, 4)

Row 3

Row 4

m807_c07_373 1/13/06 11:57 AM Page 373

The object of this game is to creategeometric figures. Each card in the deckshows a property of a geometric figure. Tocreate a figure, you must draw a polygonthat matches at least three cards in yourhand. For example, if you have the cards“quadrilateral,” “a pair of parallel sides,” and“a right angle,” you could draw a rectangle.

A complete set of rules and playing cards is available online.

Coloring TessellationsTwo of the three regular tessellations—triangles and squares—can becolored with two colors so that no two polygons that share an edgeare the same color. The third—hexagons—requires three colors.

1. Determine if each semiregular tessellation can be colored withtwo colors. If not, tell the minimum number of colors needed.

2. Try to write a rule about which tessellations can be coloredwith two colors.

Polygon RummyPolygon Rummy

KEYWORD: MT7 Games


m807_c07_374 1/19/06 4:08 PM Page 374

A

B

Project CDGeometry

PROJECT

Materials • 3 sheets of white

paper• CD or CD-ROM• scissors• tape• markers• empty CD case

It’s in the Bag! 375

C

Make your own CD to record important factsabout plane geometry.

Fold a sheet of paper in half. Place a CD on topof the paper so that it touches the folded edge.Trace around the CD. Figure A

Cut out the CD shape, being careful to leave thefolded edge attached. This will create two paperCDs that are joined together. Cut a hole in thecenter of each paper CD. Figure B

Repeat steps 1 and 2 with the other two sheetsof paper.

Tape the ends of the paper CDs together tomake a string of six CDs. Figure C

Accordion fold the CDs to make a booklet.Write the number and name of the chapter onthe top CD. Store the CD booklet in an emptyCD case.

Taking Note of the MathUse the blank pages in the CD booklet to take notes on the chapter. Be sure to include definitions and sample problems that will help you review essential concepts about plane geometry.

5

4

3

2

1

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Complete the sentences below with vocabulary words from thelist above.

1. Lines in the same plane that never meet are called ___?___.Lines that intersect at 90° angles are called ___?___.

2. A quadrilateral with 4 congruent angles is called a ___?___. A quadrilateral with 4 congruent sides is called a ___?___.

Vocabulary

Points, Lines, Planes, and Angles (pp. 324–328)7-1

E X A M P L E EXERCISES

■ Find the angle measure.

m�1m�1 � 122° � 180°

� 122° � 122°�� m�1 � 58°

Find each angle measure.

3. m�1

4. m�2

5. m�3

Stu

dy

Gu

ide:

Rev

iew

. . . . . . . . . . . . 325

. . . . . . . . . . 336

. . . . . . . . . . . . . . . . . . 325

. . . . . . 358

. . 325

. . . . . . . . . . . . . 325

. . . . . . . . 354

. . . . . 337

. . . . . . . . . . . . . . . . . 358

. . . . . . . 337

. . . . . . . . . . . . . . . . . . . 324

. . . . . . . 364

. . . . . . . . . 364

. . . . . . . . . . . 325

. . . . . . . . . 336

. . . . . . . . . . . 330

. . . . . . . . . . 342

. . . . 330

. . . . . . . . . . . . . . . . . . 324

. . . . . . . . . . . . . . . . . . 324

. . . . . . . . . . . . . . . 341

. . . . . . . . . . . . . . . . . . . . 324

. . . . . . . . . . . . . . 342

. . . . . . . . . . . . . . 358

. . . . . . . . 342

. . . . . 368

. . . . . . . . . . . . . . 342

. . . . . . . . . . . . . 325

. . . . . . . . . . 336

. . . . . . . . . . . . . . . . . . . 347

. . . . . . . . . . . . . . . 358

. . . 365

. . . . . . . . . . . . . . . . . . . 347

. . . . . . . . 337

. . . . . . . . . . . . . . . 324

. . . . . . . . . . . . . . . . . . 347

. . . . . . . . . . . . . . . . 342

. . 325

. . . . . . . . . . . . 368

. . . . . . . . 358

. . . . . . . . . . . . 358

. . . . . . . . . . . . 330

. . . . . . . . . . . . . . 342

. 336

. . . . . . . . . 325vertical angles

Triangle Sum Theorem

trapezoid

transversal

translation

transformation

tessellation

supplementary angles

square

slope

segment

scalene triangle

run

rotational symmetry

rotation

rise

right triangle

right angle

rhombus


regular polygon

reflection

rectangle

ray

polygon

point

plane

perpendicular lines

parallelogram

parallel lines

obtuse triangle

obtuse angle

line symmetry

line of symmetry

line

isosceles triangle

image


correspondence

congruent

complementary angles

center of rotation

angle

acute triangle

acute angle

1 122°2 3 1

23

68°

m807_c07_376_378 1/13/06 11:57 AM Page 376

Parallel and Perpendicular Lines (pp. 330–333)7-2


Line j⏐⏐line k. Find each angle measure.

■ m�1 m�1 � 143°

■ m�2 m�2 � 143° � 180°

� 143° � 143°�� m�2 � 37°

Line p⏐⏐line q. Find each angle measure.

6. m�1

7. m�2

8. m�3

9. m�4

10. m�5

Coordinate Geometry (pp. 347–351)7-5


■ Graph the quadrilateral with the givenvertices. Give all the names that apply.D(�2, 1), E(2, 3), F(3, 1), G(�1, �1)

D�E�⏐⏐F�G�E�F� ⏐⏐G�D�D�E� ⊥ E�F�

Graph the quadrilaterals with the givenvertices. Give all the names that apply.

14. Q(2, 0), R(�1, 1), S(3, 3), T(8, 3)

15. K(2, 3), L(3, 0), M(2, �3), N(1, 0)

16. W(2, 2), X(2, �2), Y(�1, �3), Z(�1, 1)

Stud

y Gu

ide: R

eview

143° 21

j

k

50°

n°

x

y

D

G

E

Fparallelogram,rectangle

1

p

q

2 3

4 5114°

128°

m°

m°

Angles in Triangles (pp. 336–340)7-3


■ Find n°.

n° � 50° � 90° � 180°n° � 140° � 180°

� 140° � 140°��

n° � 40°

11. Find m°.

Classifying Polygons (pp. 341–345)7-4


■ Find the angle measures in a regular 12-gon.

12x° � 180°(12 � 2)12x° � 180°(10) 12x° � 1800°

x° � 150°

Find the angle measures in each regularpolygon.

12. a regular octagon

13. a regular 11-gon

Study Guide: Review 377

m807_c07_376_378 1/13/06 11:57 AM Page 377

Stu

dy

Gu

ide:

Rev

iew

Congruence (pp. 354–357)7-6


■ Triangle ABC � triangle FDE. Find x.

x � 4 � 4� 4 � 4��

x � 8

Triangle JQZ � triangle VTZ.

17. Find x.

18. Find t.

19. Find q.

Transformations (pp. 358–361)7-7


■ Draw the image of a triangle with vertices (�2, 2),(1, 1), and (�3, �2) after a 180°rotation around (0, 0).

Draw the image of a triangle ABC withvertices (1, 1), (1, 4), and (3, 1) after eachtransformation.

20. reflection across the x-axis

21. translation 5 units left

22. 180° rotation around (0, 0)

Symmetry (pp. 364–367)7-8


■ Complete the figure. The dashed line isthe line of symmetry.

Complete each figure.

23. 6-fold 24. 25.

Tessellations (pp. 368–371)7-9


■ Create a tessellation with the figure.

Create a tessellation with each figure.

26. 27.

35

4

D

E F

x � 4A

B

C

x

y

48°7.2

(3x)°

3t

VZ

Q T

J 75°2q � 1 13


m807_c07_376_378 1/13/06 11:57 AM Page 378

Chapter 7 Test 379

In the figure, line m⏐⏐line n.

1. Name two pairs of supplementary angles.

2. Find the m�1.

3. Find the m�2.

4. Find the m�3.

5. Find the m�4.

6. Two angles in a triangle have measures of 44° and 57°. What is the measureof the third angle?

7. What are the measures of the congruent angles in an isosceles triangle if the measure of the third angle is 102°?


8. 9.

Graph the quadrilateral with the given vertices. Give all of the names thatapply to each quadrilateral.

10. A(3,4), B(8,4), C(5,0), D(0,0) 11. K(�4,0), L(�2,5), M(2,5), N(4,0)


12. rectangle PQRS with P(0,0), Q(0,4), R(4,4)

In the figure, quadrilateral ABCD � quadrilateral LMNO.

13. Find m.

14. Find n.

15. Find p.

Pentagon ABCDE has vertices A(1, �2), B(3, �1), C(7, �2), D(6, �4), and E(2, �5).Find the coordinates of the image of each point after each transformation.

16. rotation 90° around the origin, point E 17. reflection across the x-axis, point C

18. translation 6 units up, point B 19. reflection across the y-axis, point A

20. Complete the figure. The dashed line is the line of symmetry.

A

B

C

DAB || CDAD || BC

3 cm

3 cm

3 cm 3 cm Ch

apter Test

135°1 2

3 4

m

n

CD

131°

27

11

21

74°

n � 6

(m � 45)°

3p

AB

L M

O

N

m807_c07_379 1/13/06 11:58 AM Page 379

Test

Tac

kler


Extended Response Julianna bought a shirt marked down 20%.She had a coupon for an additional 20% off the sale price. Is thisthe same as getting 40% off the regular price? Explain yourreasoning.

4-point response:

The student answers the question correctly and shows all work.

3-point response:

The student makes a minor computation error that results in an incorrect answer.

2-point response:

The student makes major computation errors and does not show all work.

1-point response:

The student shows no work and has the wrong answer.

It is the same.

No, it is not the same. A $30 shirt with 20% off and then an additional

20% off is $6. A $30 shirt at 40% off is $12.

Yes, it is the same. If the shirt originally cost $25, it would cost $15

after taking 20% off of a 20% discount. A 40% discount off $20 is $15.

Shirt original price � $25

Shirt at 20% off � $20 $25 � 20% � $5; $25 � $5 � $20

Shirt at 20% off sales price � $15 $20 � 20% � $4; $20 � $4 � $15

Shirt at 40% off � $15 $25 � 40% � $10; $25 � $10 � $15

No, the prices are not the same. Suppose the shirt originally cost $40.

20% off a 20% markdown: $40 � 20% � $8; $40 � $8 � $32;

$32 � 20% � $6.40; $32 � $6.40 � $25.60

40% off: $40 � 40% � $16; $40 � $16 � $24

Scoring Rubric

4 points: The studentanswers all parts of thequestion correctly,shows all work, andprovides a complete and correct explanation.

3 points: The studentanswers all parts of thequestion, shows allwork, and provides acomplete explanationthat demonstratesunderstanding, but thestudent makes minorerrors in computation.

2 points: The studentdoes not answer allparts of the questionbut shows all work andprovides a complete andcorrect explanation forthe parts answered, orthe student correctlyanswers all parts of thequestion but does notshow all work or doesnot provide anexplanation.

1 point: The studentgives incorrect answersand shows little or nowork or explanation, orthe student does notfollow directions.

0 points: The studentgives no response.

Extended Response: Write Extended ResponsesExtended response test items often consist of multi-step problems to evaluate your understanding of a math concept. Extended response questions are scored using a 4-point scoring rubric.

m807_c07_380_381 1/13/06 11:58 AM Page 380

Test Tackler 381

To receive full credit, make sure allparts of the problem are answered. Besure to show all of your work and towrite a neat and clear explanation.

Read each test item and answer thequestions that follow.

Item AJanell has two job offers. Job A pays $500per week. Job B pays $200 per week plus15% commission on her sales. Sheexpects to make $7500 in sales permonth. Which job pays better? Explainyour reasoning.

1. A student wrote this response:

What score should the student’sresponse receive? Explain yourreasoning.

2. What additional information, if any,should the student’s response includein order to receive full credit?

3. Add to the response so that itreceives a score of 4-points.

4. How much would Janell have tomake in sales per month for job Aand job B to pay the same amount?

Item BA new MP3 player normally costs $97.99.This week, it is on sale for 15% off itsregular price. In addition to this, Jasminereceives an employee discount of 20%off the sale price. Excluding sales tax,what percent of the original price willJasmine pay for the MP3 player?

5. What information needs to beincluded in a response to receive full credit?

6. Write a response that would receivefull credit.

Item CThree houses were originally purchasedfor $125,000. After each year, the value ofeach house either increased or decreased.Which house had the least value after thethird year? What was the value of thathouse? Explain your reasoning.

7. A student wrote this response:

What score should the student’sresponse receive? Explain yourreasoning.

8. What additional information, if any,should the student’s response includein order to receive full credit?

Item DKara is trying to save $4500 to buy a usedcar. She has $3000 in an account thatearns a yearly simple interest of 5%. Willshe have enough money in her accountafter 3 years to buy a car? If not, howmuch more money will she need? Explainyour reasoning.

9. What information needs to beincluded in a response to receive full credit?

10. Write a response that would receivefull credit.

House A increased 3% over three years.

House B increased 1% over three years.

House C increased 3% over three years. So,

House B had the least value after the third

year. Its value increased 1% of $125,000, or

$1250, for a total value of $126,250.

Job A pays better.

Test Tackler

House Original Year 1 Year 2 Year 3Cost ($)

A 125,000 1% 1% 1%

B 125,000 4% �2% �1%

C 125,000 3% �2% 2%

Percent Change in Value

m807_c07_380_381 1/13/06 11:58 AM Page 381

Stan

dar

diz

ed T

est

Prep

KEYWORD: MT7 TestPrep

1. Which angle is a right angle?

�FED �GEH

�FEG �GED

2. A jeweler buys a diamond for $68 andresells it for $298. What is the percentincrease to the nearest percent?

3% 138%

33% 338%

3. A grocery store sells one dozen ears ofwhite corn for $2.40. What is the unitprice for one ear of corn?

0.05/ear of corn

$0.20/ear of corn

$1.30/ear of corn

$2.40/ear of corn

4. The people of Ireland drink the mostmilk in the world. All together, theydrink more than 602,000,000 quartseach year. What is this number writtenin scientific notation?

60.2 � 105

602 � 106

6.02 � 108

6.02 � 109

5. Cara is making a model of a car that is14 feet long. What other information is needed to find the length of themodel?

Car’s width Scale factor

Car’s speed Car’s height

6. For which equation is the point asolution to the equation?

y � 2x � 1 y � �x � 1

y � 2x � 2 y � �2x � 2

7. What is q in the acute triangle?

62 118

72 128

8. Which expression represents “twice thedifference of a number and 5”?

2(x � 5) 2(x � 5)

2x � 5 2x � 5

9. For which equation is x � �1 the solution?

3x � 8 � 11 �3x � 8 � 5

8 � x � 9 8 � x � 9DB

CA

JG

HF

DB

CA

JG

HF

DB

CA

J

H

G

F

D

C

B

A

JG

HF

DB

CA

Cumulative Assessment, Chapters 1–7Multiple Choice


F

E

D

H

G

70°

20°

O 42�2

�2

�4

�4

x

y

80° 38°

q°

m807_c07_382_383 1/13/06 11:58 AM Page 382

Cumulative Assessment, Chapters 1–7 383

10. Marcus bought a shirt that was on salefor 20% off its regular price. If Marcuspaid $20 for the shirt, what what itsregular price?

$25 $16

$40 $30

Gridded Response

Use the following figure for items 11 and12. Line p is parallel to line q.

11. What is the measure of �4, in degrees?

12. What is the sum of the measures of �2and �6, in degrees?

13. Maryann bought a purse on sale for25% off. She paid $36 for the pursebefore tax. How much did the pursecost originally?

14. What is the value of the expression�2xy � y2, when x � �1 and y � 4?

15. A parallelogram has vertices at A(�2, 4), B(�1, �1), C(1, 0), and D(0, 5). What is the x-coordinate of Bafter the parallelogram is reflectedover the y-axis?

16. Guillermo invests $180 at a 4% simpleinterest rate for 6 months. How muchmoney will Guillermo earn in interest?Write your answer as a decimal to thenearest tenth.

Short Response

17. Triangle ABC, with vertices A(2, 3), B(4, �5), C(6, 8), is reflected across the x-axis to form triangle A�B�C�.

a. On a coordinate grid, draw andlabel triangle ABC and triangleA�B�C�.

b. Give the new coordinates fortriangle A�B�C�.

18. Complete the table to show thenumber of diagonals for the polygonswith the numbers of sides listed.

Extended Response

19. Four people are introduced to eachother at a party, and they all shakehands.

a. Explain in words how the diagramcan be used to determine thenumber of handshakes exchangedat the party.

b. How many handshakes areexchanged?

c. Suppose that six people wereintroduced to each other at a party.Draw a diagram similar to the oneshown that could be used todetermine the number ofhandshakes exchanged.

JG

HF

Stand

ardized

Test Prep

Use logic to eliminate answer choicesthat are incorrect. This will help you tomake an educated guess if you arehaving trouble with the question.

Number of Sides Number of Diagonals

3 0

4

5

6

7

n

1

p

q

23

45

67

130°

BA

DC

m807_c07_382_383 1/13/06 11:58 AM Page 383

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