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Geometry 1st Semester Review 2012 FCS, 2012-13, Mr. Garcia

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12/11/12 5:47 PM Name_________________________________ Date___________________ Period______ This is your semester exam which is worth 10% of your semester grade. You can determine grade “what-ifs” by using the equation below. (Current Re nWeb Grade)x .90 + ( final exam grade) x .10 = semester grade The first semester exam will cover geometry content from chapters 1 to 5. A calculator will be allowed. Most of the major formulas will be provided. ======================================================================== A. Chapter 1: Points, Lines, and Planes

1. Terms:

Point, line, plane, segment, ray, angle, angles, vertex, slope of a line

y2 − y1

x2 − x1

, space, collinear, noncollinear, coplanar,

noncoplanar, distance 2 2a b+ or (x2 − x1)2 + ( y2 − y1)2 , midpoint ' '( , )

2 2sum of x s sum of y s

or

1 2 1 2( , )2 2

x x y y+ +, congruent segments (are =in length), Segment addition propery.

2. Practice Problems Refer to the figure to the right to answer problems 1 - 10.

______1. The line intersecting plane P.

______2. The intersection of AC and XF .

______3. Are points B, F, and X collinear?

______4. Are points A, B, and X coplanar?

______5. Are points A, B, and X contained in Plane P?

____ ____ ____6. Identify 3 non-collinear points

____ ____ ____ ____7. Identify 4 non-coplanar points. ______, ______ 8. Identify 2 angles that have B as their vertex. ______9. Name a line. ______10. Name a ray.

3. Segment and midpoint definitions

• 2 segments having the same length are ≅ . • The midpoint of a segment is a point half way between the endpoints of the segment. Thus, if X

is the midpoint of AB , then AX = XB, also 2AX = AB, or 2XB = AB.

j

X

C B A

P D F


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• A Segment bisector is any segment, line, ray, or plane that intersects a segment at its midpoint. • The Addition Postulate says that if point C is between the endpoints A and B of a segment, then

AC + CB = AB.

4. Practice problems concerning segments and midpoints: 12. If B is the midpoint of AC and AB = 2x – 3 and BC = 5x – 24, find x, AB, and BC. x = _____, AB = _____, BC = _____ 13. If XB = 14 and XF = 20, find BF. BF = ______ 14. If B is the midpoint of XF and XB = x + 11 and BF = 5x – 1, find x and XF. x = ____ , XF = ______ 15. If AB = 3x, BC = x + 2, and AC = 38, find x and AB. x = _____, AB = _____

16. If the x coordinate of G is –8 and the x coordinate of H is 9, find GH. GH = ______ 17. Find the midpoint of the segment having the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ 18. Find the distance between the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ d. If the length of PQ is twice the length of AB, then find PQ. _____ e. If the length of RS is one third the length of EF, then find RS. _____ 19. How many sides does a pentagon have? ______ 20. What does it mean for a polygon to regular? ____________________ 21. Find the perimeter of a regular hexagon with a side equal to 5. ______

=========================================================================================== B. Chapter 2 Topics: Logic & Reasoning

1. Terms: Deductive reasoning, inductive reasoning, conjecture, conditional, hypothesis, conclusion, converse, contrapositive, counterexample 2. Practice Problems A. Restate each of the following given statement into an “if-then” statement. B. Underline the hypothesis and circle the conclusion. C. Is the statement true or false? Circle your answer. D. Write the converse of the conditional and determine whether it is true or false. E. Write the inverse of the conditional and determine whether it is true or false. F. Write the contrapositive of the conditional and determine whether it is true or false. G. If possible, write the bi-conditional statement in “if and only if” form. If not, write a counter example

demonstrating why not.


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1. Tardy students receive detention. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ 2. All right angles are congruent. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ 3. A triangle is a polygon that has three sides. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ 4. Supplementary angles are two angles whose sum is 180°. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F

G. ___________________________________________________________________ ============================================================================ C. Chapter 3 Topics: Angle Relationships

1. Terms: The vertex angle & sides of an angle, naming an angle ( BDG∠ means point D is the vertex of the angle), right (= 90º), acute (< 90º ) and obtuse angles (> 90º ), congruent angles(≅ ) adjacent angles, vertical angles, complementary angles(sum is 90º), supplementary angles (sum is 180º), linear pair(sum is 180º), perpendicular lines (Lines that form 90º angles), slope of parallel lines are equal, slope of

perpendicular lines are “opposite reciprocals” 2 3( , )3 2

If m then m⊥= = −

2. Practice Problems Refer to Figure 2. Matching, you may use more than one letter to describe the angle(s). ________ 1. ∠1 and ∠2 ________ 2. ∠1 and ∠5 ________ 3. ∠3 and ∠4 ________ 4. ∠1 and ∠BOE ________ 5. ∠1 and ∠6 ________ 6. ∠AOF and ∠BOE ________ 7. ∠AOC and ∠COE ________ 8. ∠2 and ∠5 ________ 9. ∠4 and ∠AOD

a. acute angles b. right angles c. obtuse angles d. adjacent angles e. linear pair f. complementary angles g. supplementary angles h. vertical angles i. congruent angles

Figure 2

A

B C D

E

F G

1

2 3 4

5 6 O •

•

• •

•

• •

•


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Refer to figure 2 to solve problems 10 - 17. 10. If m∠3 = 27°, then m∠4 = _____ 11. m∠1 + m∠BOD = m∠_____. 12. If m∠1 = 46° and m∠4 = 59°, then m∠DOF = _____. 13. If OD bisects ∠COE, then m∠4 = _____. 14. If OD ⊥ BF , then m∠4 + m∠5 = _____. 15. If OD ⊥ BF and m∠4 = 65°, then m∠1 = _____, m∠2 = _____, m∠6 = _____, m∠AOF = _____. 16. If OD ⊥ BF , name all the pairs of complementary angles.______________ _____________________________________________________________ 17. If OD is the ⊥ bisector of BF , which segments are congruent? __________ Refer to figure 3 to solve problems 18 - 21. 18. Given: m∠2 = 9x +28 and m∠3 = 47 – 2x, x = _____, m∠2 = _____ 19. Given: m∠1 = 3x + 5 and m∠3 = 65, x = _____ 20. Given: m∠2 = 9x +2 and m∠4 = 7x + 36, x = _____, m∠2 = _____ 21. Given: m∠1 = x-9 and m∠2 = 2x, x = _____, m∠1 = _____

D. Chapter 3 Topics: parallel Lines & Their Relationships 1. Terms: Parallel (//) lines, transversal, corresponding angles (≅ ), alternate interior angles (≅ ), alternate exterior angles (≅ ), same side (or consecutive) interior angles (sum of 180), (supplementary angles still occur), parallel lines never intersect, parallel lines have the same slope 2. Practice Problems For problems 1-14, refer to figure 4 to determine which lines if any are parallel. 1. Given: ∠1 ≅ ∠5 _____ 2. Given: ∠8 ≅ ∠12 _____ 3. Given: ∠7 ≅ ∠13 _____ 4. Given: ∠4 ≅ ∠14 _____ 5. Given: ∠6 ≅ ∠11 _____ 6. Given: ∠10 ≅ ∠15 _____ 7. Given: ∠3 and ∠13 are supplementary _____ Given a b, l m . (Refer to figure 4) 8. If m∠ 12 = 67o , then m∠ 3 = ______ 9. If m∠ 6 = 108o, then m∠ 16 = ______ 10. If m∠ 4 = 123o, then m∠ 10 = ______ 11. If m∠ 1 = 71o, then m∠ 10 = ______ 12. m∠1 = 2x + 7 and m∠16 = x + 30, x = _____, m∠1 = _____, m∠16 = _____ 13. m∠11 = 3x + 6 and m∠13 = x + 26, x = _____, m∠11 = _____, m∠13 = _____ 14. m∠2 = 11x - 16 and m∠7 = 7x + 28, x = _____, m∠2 = _____, m∠7 = _____

Find the slope of the line through the given points.

15. a. A(-3,8), B(4,2) _____ b. What is the slope of any line parallel to the line through points A and B? _____ c. What is the slope of any line perpendicular to the line through points A and B? _____

Figure 2

A

B C D

E

F G

1

2 3 4

5 6 O •

•

• •

•

• •

•

1 2

3 4

Figure 3

1 2 3

4

5 6

7 8 9 10

11 12

13

14 15

16

a b

l

m

Figure 4


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16. a. C(1,-3), D(9,-9) _____

b. What is the slope of any line parallel to the line through points C and D? _____ c. What is the slope of any line perpendicular to the line through points C and D? _____

17. a. E(-2,-3), F(-6,-5) _____

b. What is the slope of any line parallel to the line through points E and F? _____ c. What is the slope of any line perpendicular to the line through points E and F? _____

============================================================================= E. Chapter 4 Triangle Relationships 1. Term: {classified by angles} right (1 right Δ ), acute (all acute Δ ’s), obtuse(1 obtuse Δ ), equiangular triangles (all 60° angles). {Classified by sides}, Scalene (no sides are =), isosceles (at least 2 sides are =), equilateral triangles (all sides are =) .sum of the interior angles is 180°, sum of the remote interior angles is = to the exterior angle of the triangle, 2. Practice Problems Find the value of x. 1. x =_______ 2. x = _______ 3. x = _______ In ΔABC, find x and m∠A, then classify the type of triangle according to sides and angels. 4. 246 −=∠ xAm , 72 −=∠ xBm , and 4+=∠ xCm x = ______, =∠Am Classification:_by sides:___________________ by angles: ____________________ 5. m∠A = 8x + 9, m∠B = 3x – 4, m∠C = 9x + 15 x = ______, =∠Am Classification:_by sides:___________________ by angles: ____________________ Using the given information, classify each triangle according to its sides and angles. 6. DFZΔ , DF < DZ and m∠ D = 90. 10. MNOΔ , 27=∠Mm and 82=∠Om .

7. AWVΔ , AW = AV and m∠A < 90. _________11. LJRΔ , 35=∠Lm and 104=∠Rm .

8. PONΔ , PO = 5, ON = 5, PN = 5. 12. KMNΔ , Mm∠ >90°, MN = MK.

9. LJIΔ , 45=∠Lm and 90=∠Im . 13. SYXΔ , Sm∠ = 60° and Ym∠ = 60°.

Use the distance formula to classify the triangle by the measure of its sides. 14. A(1, 0) B(3, 3) C(2, 4) AB = _____ BC = _____ AC = _________ Classification: ______________

100°

x° x

70°

x

70°


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15. D(4, -6) A(-2, 5) V(0, 7) DA =_____ AV = _____ DV = Classification: _______________ F. Chapter 4 Topic: Congruent Triangles 1. Term: constructions of congruent triangles, 2 sides and the included angle are ≅ (SAS), 2 angles and the included side are ≅ (ASA), three congruent sides are ≅ (SSS), 2 angles and the non-included side are ≅ (AAS). 2. Practice Problems Identify the congruent triangles and justify your answer. If congruency can not be proven write “n p” in both blanks. 1. Given: FDCAandEFBCEDAB ≅≅≅ ,, ΔBAC ≅ Δ__________ by _____________.

2. Given: MPandMPMPMTSM ,, ≅≅ bisects ∠SMT

ΔMPS ≅Δ__________ by _____________. 3. Given: ,MNOM ⊥ ,PQPR⊥ ,PRMO ≅ and RQON ≅

ΔMNO ≅ Δ_________ by ______________.

4. Given: HKGHJKFG ≅≅ , ΔHJK ≅ Δ_________ by _______________.

5. 6. For the following problems, ΔABC ≅ ΔDEF. 7. Given: AB = 3y + 12, DE = 5y – 18, find DE. ______ 8. Given: m∠C = 4y – 23, m ∠F = 2y – 5, find the m∠C. ______ 9. Use the distance formula to determine whether the triangles with the given vertices are congruent. Given: ∆PQR : P(1,2), Q(3,6), R(6,5) ∆ KLM : K(-2,1), L(-6,3), M(-5,6) PQ = KL = QR = LM = PR = KM = Are they Congruent? Why?

M

S P T

M

P

O

R

Q

N F

J

H

G

K

A B

C

D

F

E

Given: C is the Midpoint of ΔABC ≅ Δ _______by _________________

A

C

E

B

D Given: bisects ∠YXW, ∠YZX is a right angle. ΔXYZ ≅ Δ ________ by ________________

Y Z

W

X


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Proofs: 1. Given : a // b and m // n

Prove: ∠ 4 ≅ ∠ 10

Statements Reasons

1.

2.

3.

4.

2. Given : AD // BC ; AD ≅ BC

Prove: ∆ ABD ≅ ∆ CDB Statements Reasons

1. 2. 3. 4.

1. Describe the location of point E. Point A. Point D. 2. Where does to triangle formed by points B, E, and C lie? 3. Where does the line containing points B and C lie? 4. Does a line containing point E have to intersect the plane? 5. If GHKΔ is a right triangle, name another right triangle. 7. Name a pair of alternate interior angles. 8. Name a pair of corresponding angles.

10 14

A B

C D 1

2

3 4

K

L

M

F

G H

J

I

•

•

• •

P

O

A D

C B

E

M

6 5 4 3

a

2 1

11

8 7

16 15 12

n m

b

9 10 14


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9. Use the graph to the right and use the Pythagorean Theorem to determine the length of the longest segment. Round to the nearest hundredth. Be sure to indicate the segment. List the segments is order from least to greatest. 10. Use the graph to the right to answer the following questions. State the coordinates for an endpoint of the segment with point B as one endpoint and point A as a midpoint. 11. Given: C is the midpoint of BD , BC = (2x – 3) cm and CD = (5x – 24)cm. Find the length of BD . 12. Find the value of x in the figure. 13. If m∠FBC = 74 and m∠ 1 = 3x - 8 and m∠ 2 = 5x + 26, find x and m∠ 3.(4 points)

14. If m∠ 1 = °41 , and m∠ DOF = °87 , what is m∠ 4? 15. If m∠ 3 = 8x – 12 and m∠ 4 = 4x + 6, and m∠ 1 = 3x – 9, find m∠ 1. 16. If ∠ 3≅ ∠ 4, then OD

is a(n) ______________.

17. If GA

// BF

, their slopes are _______. 18. If point B and point D are equidistant from AE

, what conclusion

can be made about ∠ 1 and ∠ 4? 19. What is the sum of ∠ 1, ∠2, ∠3, ∠4, ∠5, and ∠6?

If the equation of line 1 is 13 ( 5)3

y x+ = − + , state a possible equation which would describe line 2.

20. KNGΔ is an isosceles triangle with K∠ as the vertex angle, and 5 2KN x= − , and 2 4GK x= + .

a. Draw a diagram and label the angles and the sides with their lengths in algebraic form.

b. What is the length of KN ? …Of KG ? c. For what range of values for GN will the lengths still form a triangle ? d. Make a table of lengths possible for NG . (Use only integers) e. Using the range of values above, find 1 value that will form an ACUTE triangle. Justify using the Pythagorean

theorem. f. Using the range of values above, find 1 value that will form an OBTUSE triangle. Justify using the Pythagorean

theorem.

A

B

E

F G

1

Figure 2

C D

2 3 4

5 6 O •

•

• •

•

• •

•

A

F

C

D

B 1

2 3

3x + 4

2x + 1

•B

•C

•F

•D • A


9

Why are the following triangles are congruent? Justify your reasoning! Be sure to use the phrase “two sides and the included angle are congruent” instead of SAS! 21. Given : AB ≅ CD; AD ≅ BC 22. Given: AE ≅ BC ; ∠E ≅ ∠C, D is the midpoint of Prove : ∆ABD ≅ ∆ CDB Prove: ∆ADE ≅ ∆BDC 23. Determine which postulate can be used to prove the triangles are congruent. If the triangles cannot be proven congruent write NONE. Be sure to write out the postulate (EX: 2 sides and the included angle are congruent instead of SAS)

=============================================================================================== G. Chapter 5 Triangle Relationships 1. Terms: altitude, centroid, circumcenter, concurrent lines, incenter, median, orthocenter, perpendicular bisector, point of concurrency, perpendicular bisector, angle bisector. 2. Chapter 5 Theorems: 5.1 ⊥ If a point on a ⊥ bisector of a segment, then it is equidistant from the endpoints of the segment. 5.2 Converse of the ⊥ bisector theorem is also true. 5.3. Circumcenter theorem: The ⊥ bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle. 5.4 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 5.5 The converse of the angle bisector theorem is also true.

EC

A B

C D A B

E D C


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5.6 Incenter Theorem The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. The point of concurrency of the angle bisectors is called an incenter. 5.7 Centroid Theorem The medians of a triangle intersect at a point called the centroid that is 2/3 of the distance from each vertex to the midpoint of the opposite side. The point of concurrency of the medians is called the centroid. An altitude of a triangle is a segment from a vertex to the line containing the opposite side and ⊥ the line containing that side. The lines containing the altitudes of a triangle are concurrent, intersecting at a point called the orthocenter. Concurrent lines are the 3 ⊥ bisectors, or the 3 angle bisectors, or the 3 medians, or the 3 altitudes drawn in a triangle. The altitude, median, ⊥ bisector, or the angle bisector 2. Practice Problems:

a. Make a drawing of each of the theorems above, without looking at your book!

b.


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Complete the following. Show all work. 1. Given: BF is the ⊥ bisector of AD m∠AEF = 150 m∠BDE = 80

Find: m∠1 = ____________

m∠2 = ____________

m∠3 = ____________

m∠4 = ____________

2. Given: BD is the ⊥ bisector of AC m∠ABE = 40 m∠ECD = 70

Find: m∠AED = ____________

m∠ABC= ____________

m∠ADE = ____________

3. Given: AD is the ⊥ bisector of EF m∠AEB = 35 m∠AFC = 80 m∠FDC= 15

Find: m∠1 = ____________

m∠2 = ____________

m∠3 = ____________

m∠4 = ____________

3. Inequalities in One Triangle Definition of inequality: For any real numbers a and b, a > b, iff, there is a positive number c such that a = b + c. 5.8 Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle > the measure of either of its corresponding remote interior angles. Angle side relationships in triangles theorems: 5.9 If one side of a triangle is longer than another side, then the angle opposite the longer side has greater measure than the angle opposite the shorter side. 5.10 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. 1. In ΔQRT, the angles listed from largest to smallest are:

a) ∠ Q , ∠R , ∠T b) ∠ R , ∠Q , ∠T

c) ∠ T , ∠R , ∠Q d) ∠ Q , ∠T , ∠R

A

B C

D

E F • 1

2

3 4

B

A

E

C

D

A

E

F

C B D 1

2 3

4

T R

Q

25 19

30


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4. The triangle Inequality 5.11 Triangle Inequality Theorem The sum of the lengths of any 2 sides of a triangle must be > than the length of the 3rd side. Practice problems: Is it possible to form a triangle with the given lengths? If not, explain why not. 1) 5, 6, 9 2) 3, 4, 8 Find the range for the measure of the 3rd side of a triangle given the measure of the 2 sides: 3) 5 ft., 7 ft. 4) 10.5 cm, 4 cm

current renwebgrade x.90 finalexamgrade .10 … · point, line, plane, segment, ray, angle, angles,...

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