Geometry Final Review Guide Name:_________________________

District 2013

1. Name all segments shown.

X Y Z

2. Describe the figure below.

A B

3. Describe the figure below.

C D

4. Name all rays shown.

XY Z

5. If AB = 16 and AC = 22, find the length of BC.

A B C

6. Which one of the following gives the distance between the numbers 7 and –2?

[A] –5 [B] 5 [C] 9 [D] –9

7. SQ�

bisects ∠ RST. Find m∠ RSQ if m RST∠ = 48.

S

T

Q

R

8. Solve: In the figure (not drawn to scale), MO�

bisects ∠ LMN, m∠ LMO = 12 33x − , andm NMO∠ = x + 44. Solve for x and find m∠ LMN.

OL

M

N

[A] 1, 40 [B] 7, 102 [C] 7, 117 [D] 1, 21

9. Find the coordinates of the midpoint of MN with endpoints M = (2, 3) and N = (2, 9).

10. Find the coordinates of B if A is the point (−1, 4) and the midpoint of AB is M = (5, 4).

11. The measures of two angles of a triangle are 23° and 35° . Find the measure of the thirdangle.

12. Solve for x:

5 + 21x

5 + 129x

13. If ∠ A and ∠ B are complementary angles and m A m B∠ = ∠4 , find m A∠ and m B∠ .

14. In the figure shown, m∠ AED = 114. Which of the following statements is false?

A D

B C

E

[A] m ∠ AEB = 66 [B] m ∠ BEC = 114

[C] ∠ BEC and ∠ CED are vertical angles.

[D] ∠ AEB and ∠ DEC are congruent angles.

15. Name two vertical angles, two supplementary angles, and two adjacent angles in thediagram below.

34

56

1

2

16. Find the distance between points P( − 5, 4) and Q( − 3, 1).

17. Find the distance between points P(2, 3) and Q( − 4 , 5).

18. Find the value of x. (The triangle is not drawn to scale.)

x

17°

19. Find the value of x. (The triangle is not drawn to scale.)

x

119°

28°

20. Find the value of x.

119°

61°

x°

21. Find the values of x, y, and z.

64° x° y°

29°

z°

21°

22. Find the values of x, y, and z.

65° x° y°

26°

z°

31°

23. A ladder is against a building. The angle it makes with the ground is twice the angle itmakes with the building. How large is the exterior angle formed between the ground andthe ladder?

24. Find the pair of parallel lines.(1) − +8 10y x = –1(2) 10 8x y+ = –1(3) 4y = − −5 1x

25. Determine whether the graph of y x = − + 3 and 4 4 7x y+ = are perpendicular lines.

26. Given: AB is the diameter of circle O and mBC⌢

= 52. Find mCAB⌢

.

OA B

C

27. Given: AB is the diameter of circle O and mBC⌢

= 46. Find mCAB⌢

.

OA B

C

28. Find the coordinates of the center and the radius of the circle with diameter CD.C(1, –1), D(–7, 3)

29. Find the coordinates of the center and the radius of the circle with diameter CD.C(2, 3), D(–6, 7)

30. Given: ∆ ABC ∼ ∆ DEF, solve for x and y.

A

B C

D

E F

yx

610

189

31. In the figure shown, PQ = 20 centimeters, ST = 10 centimeters and m∠ QRP = 69. Findm S∠ .

P Q

T S

R

32. In the figure shown, PQ = 20 centimeters, ST = 10 centimeters and m∠ QRP = 63. Findm S∠ .

P Q

T S

R

[A] 54 [B] 27 [C] 117 [D] 63

33. In the figure shown, ∆ ABC ∼ ∆ ADE, AB = 2 yards, BC = 9 yards, AE = 18 yards, andDE = 18 yards. Find CE.

D E

A

CB

34. In the figure shown, ∆ ABC ∼ ∆ ADE, AB = 7 yards, BC = 9 yards, AE = 20 yards, andDE = 36 yards. Find BD.

D E

A

CB

35. Graph the triangle whose vertices have the coordinates given below. Then draw itsreflection in the x-axis.M(–8, 3), N(–2, 3), P(–3, 6)

36. Draw the image produced when ∆ ABC is reflected in the y -axis.

x

y

–6 6

–6

6

A

BC

37. Draw the image produced when ∆ ABC is translated 3 7, .

x

y

–10 10

–10

10

A B

C

38. What is the translation image of ( , 9)−1 under the translation ( , )x y x y ( , )→ + −2 1 ?

39. At the half-time show, a marching band marched in formation. The lead drummer startedat a point with coordinates ( , – )3 5 and moved 3 steps up, and 4 steps right. What are thecoordinates of the drummer's final position?

40. Describe the translation of point P(2, 3) to point P ′ (4, 7).

41. Find the coordinates of the image of a triangle with vertices A( , ),0 8 B(– , ),5 0 andC(– , )7 2 under a rotation of 90° clockwise about the origin.

42. Rectangle PQRS is rotated 90° clockwise about point O. Find the image of RS.

A B

Q R

SP

D C

O

y

x

[A] PQ [B] CD [C] AD [D] PS

43. Write a two-column proof of the following.Given: BD CE =Prove: BC DE =

B C D E

44. Given: SQ bisects ∠ RST . Find QR if UT = 15 and UQ = 20. (not drawn to scale)

T

S

Q

R

U

45. Given: SQ bisects ∠ RST . Find QR if UT = 40 and UQ = 75. (not drawn to scale)

T

S

Q

R

U

46. Find the length of the leg of this right triangle. Leave the answer in simplist radical form.

a

17

20

47. Use the Pythagorean theorem to solve for x.

x

207

48. You are planting and staking trees for a landscaper. The stakes must be 2 ft from the baseand the wires must extend 5 ft up the tree. How long will the wires be? Leave the answerin radical form.

49. Given AB AD and BE DE: ,≅ ≅Prove: ∆ ∆ABE ADE≅

A

D B C

E

50. Find tan P.

21

20 29

P

Q

51. Find tan P.

15

8 17

P

Q

52. A photographer shines a camera light at a particular painting forming an angle of 41°with the camera platform. If the light is 50 feet from the wall with the painting, how highabove the platform is the painting?

50 ft

x

41°

[A] cos( )4150

° = x[B] sin( )41

50° =x

[C] tan( )4150° =x

[D] tan( )4150

° = x

53. Find cos A for the right triangle below:

A C

B

12

513

54. Find sin P.

12

513

P

55. A man 6 ft tall walks 75 ft from the base of a tree. He uses a protractor to measure theangle from his eye to the top of the tree. He finds it to be about 25�. Find the height ofthe tree to the nearest foot.

56. If m∠ RST = 68, what is m∠ RQT?

QS

R

T

57. Find the diameter of the circle (not drawn to scale). BC = 11, and DC = 22. Roundyour answer to the nearest tenth.

O

A

B

CD

58. To find the radius of a circular pond, Maria measured CD and BC. If CD is a tangent,find the radius of the pond.

A

12 ydDC

B 4 yd

59. If m∠ RST = 66, what is m∠ RQT?

QS

R

T

60. If mDE⌢

= 117 and mBC⌢

= 69, find m∠ A.

D

E

B

C

A

61. If mDE⌢

= 123 and mBC⌢

= 87, find m∠ A.

D

E

B

C

A

62. A tree 16 feet tall casts a shadow which forms an angle of 40° with the ground. Howlong is the shadow?

x

16 ft

40°

63. In kite ABCD, prove AC bisects BD.

A

B

C

D

64. Find the sum of the interior angles of a regular octagon.

65. Find the measure of each interior angle of a regular hexagon.

66. If ∆ ABC ≅ ∆ DEF, AB = 5 cm, ∠ = °ABC 21 , and ∠ = °DFE 69 , which of thefollowing statements is false?

[A] ED = cm5 [B] ∠ °CAB = 90 [C] AC DF≅ [D] ∠ ≅ ∠B D

67. If ∆ ABC ≅ ∆ DEF, AB = 4 cm, ∠ = °ABC 11 , and ∠ = °DFE 29 , which of thefollowing statements is false?

[A] ∠ ≅ ∠D A [B] FD = cm4 [C] ∠ °CAB = 140 [D] BC EF≅

68. Graph the triangle whose vertices have the coordinates given below. Then draw itsreflection in the x-axis.M(–6, 2), N(–2, 2), P(–3, 7)

69. What are the coordinates of the reflection image of P( − 5, − 9 ) in the x -axis?

70. A point P has coordinates (1, –6). What are its new coordinates after reflecting point P inthe y-axis?

71. Describe two different isometries under which ∆DEF is an image of ∆ABC.

x

yA

B

C

D

E

F

0

72. The circumference of a circle is 68π cm. Find the diameter, the radius, and the length of

an arc of 50�.

73. For a circle of radius 5 feet, find the arc length of a central angle of 18° . Leave youranswer in terms of π .

74. For a circle of radius 4 feet, find the arc length of a central angle of 30° . Leave youranswer in terms of π .

75. Find the area and circumference of the circle in terms of π .

28

76. Find the area of the shaded region.

8

40°

77. What is the area of a circle whose diameter is 14 centimeters?

78. Find the volume of the cylinder. Use 3.14 for π .

4 ft

7 ft

79. Find the volume of the cylinder. Use 3.14 for π .

10 ft

14 ft

80. Estimate the volume of a cone with height 8 feet and radius 4 feet.

81. Calculate the volume of the pyramid.

17 m

17 m

15 m

82. Calculate the volume of the pyramid.

10 m

9 m

8 m

83. Find the volume of the cone.

4 m

3 m

84. Find the volume of the sphere. (Use 3.14 for π .)

4.3 in.

85. The volume of the following napkin ring, in cubic millimeters, is

10 mm

10 mm16 mm

86. Identify ∠ 3 and ∠ 6 as corresponding angles, alternate exterior angles, alternate interiorangles, same-side interior angles, or none of these.

1 2

34

5 6

78

87. Identify ∠ 3 and ∠ 7 as corresponding angles, alternate exterior angles, alternate interiorangles, same-side interior angles, or none of these.

1 2

34

5 6

78

88. In the figure shown HC GD← → ← →

|| and m∠ABC = 137. Which of the following statements isfalse?

D

G

C

H

A

F

E

B

[A] ∠ABH and ∠AEG are corresponding angles. [B]m ∠ =GEF 43

[C] m ∠ =DEF 43 [D] ∠HBF and ∠AED are alternate interior angles.

89. In the figure shown HC GD← → ← →

|| and m∠ABC = 146. Which of the following statements isfalse?

D

G

C

H

A

F

E

B

[A] ∠ABH and ∠AEG are corresponding angles.

[B] ∠HBF and ∠AED are alternate exterior angles.

[C] m ∠ =GEF 146 [D] m ∠ =DEF 34

90. In the figure shown HC GD← → ← →

|| and m∠ABC = 142. True or false:∠HBF and ∠AED are alternate interior angles.

D

G

C

H

A

F

E

B

91. Line r is parallel to line t. Find m∠ 6 and m∠ 4.

7

1 3

5 6

4 2

r

t

135°

92. Find m∠ 1 in the figure below. PQ�

and RS�

are parallel.

QP

SR 1

71°

93. What must be the value of x for a to be parallel to b?

a

b

(–4x + 5)°

40°

94. What must be the value of x for a to be parallel to b? m∠ 1 = 101 and m∠ 2 = 7 12x +

a

b

1

2

95. AC DC≅ and BC CE≅ . Prove ∆ ∆ABC DEC≅ .

B

A

CD

E

96. Find the measures of the numbered angles in the parallelogram.

34° 120°

12

3

97. Refer to the figure below. Give a congruence statement for two triangles in the figure andname the theorem or postulate that proves the congruence.I

H

J

K

HI KJ~=IK JH~=

98. Refer to the figure shown. Give a congruence statement for two triangles in the pictureand name the theorem or postulate that proves the congruence.

B C

D

E FBC = FEBC EF||

99. Refer to the figure shown. Give a congruence statement for two triangles in the pictureand name the theorem or postulate that proves the congruence.

~=

H

I

J

K

L

H L

HJ = JL

100. Refer to the figure shown. Give a congruence statement for two triangles in the pictureand name the theorem or postulate that proves the congruence.

U

TV W

XTV WV UV XV~= ~=

101. Find the perimeter of ∆ DEF if ∆ ABC ∼ ∆ DEF.

A B

C

D E

F

6

12

15 8

102. The triangles below are similar. Find the length of x.

4

10

12

45

18

x

103. The triangles below are similar. Find the length of x.

3

10

11

35

10.5

x

104. Which of the following transformations creates a figure that is similar (but not congruent)to the original figure?I. translation II. rotation III. dilation