Geometry Final Exam - Study Guide

1. True or False In Euclidean geometry, there is exactly one line through any three points.

2. M, N, O, and P are points on the number line shown. If N has coordinate –1.8, P has coordinate 4.0, NO = 3.6,

and PM = 6.5, find MN.

3. Refer to the network shown.

a. How many even nodes are there?

b. Is the network traversable?

4. Suppose Q, R, and S are points in a plane. If QR = 4, QS = 10, and RS = 6, is it possible for one of the three

points to be between the other two? Explain your answer.

5. Tell whether each set of numbers could be the lengths of the three sides of a triangle.

a. 4, 5, 9

b. 6, 6, 11

c. , 1, 1

____ 6. Which type of geometry best describes a line as an arc connecting either two nodes or one node to itself?

A graph theory

B discrete geometry

C Euclidean geometry

7. Match each term with the most appropriate drawing.

a. nonconvex hexagon

b. convex pentagon

I. II.

III. IV.

8. Use the definition of a polygon to explain why the figure shown is NOT a polygon.

Consider the conditional statement: If a number is divisible by 3, then it is divisible by 9.

9. a. Give an instance of the conditional.

b. Is the conditional true or false? If it is false, give a counterexample.

10. a. Write the converse of the conditional.

b. Is the converse true or false? If it is false, give a counterexample.

11. Joanne has three friends with red hair. All of them are good at math. Joanne conjectures, “All people with red

hair are good at math.”

a. Is Joanne’s conjecture true or false?

b. How would you show that your answer to Part a is correct?

12. Place in a correct hierarchy: polygon, triangle, quadrilateral, scalene triangle, equilateral triangle, isosceles

triangle.

13. Use the figure shown to name:

a. a pair of vertical angles.

b. an angle adjacent to .

c. an angle that appears acute.

d. a straight angle.

e. an angle that appears obtuse.

14. and are supplementary. If , then ____.

15. Use the diagram shown. bisects , , and . Find .

16. V is the center of the circle shown. If , find

a. .

b. .

c. .

17. Which lines are parallel in the figure shown?

18. Given that and name and write the statement of the theorem or postulate that justifies the

conclusion that .

19. Consider the line 2x + 7y = 42.

a. Give the slope of the line.

b. Give the slope of a line parallel to the given line.

c. Give the slope of a line perpendicular to the given line.

Consider the quadrilateral FOUR where F = (–5, 3), O = (1, 0), U = (0, –5), and R = (–5, –4).

20. Draw and label FOUR and . Give the coordinates for .

21. True or False

a.

b.

c.

d.

22. Draw and label the reflection of points P, Q, and R over line m.

23. Draw the line over which the letter Z is reflected onto its image.

24. On the miniature golf hole shown, draw a path that ball B can take to roll into hole H in one shot.

A, B, and C are three noncollinear points. None of them is on line p, and .

Determine whether each statement is true or false. If the statement is true, give a reason to justify it. If

the statement is false, draw a counterexample.

25.

26.

27.

28. Name three properties that are preserved under a translation.

29. Name an isometry that does not preserve orientation.

30. Draw two lines so that the composite of reflections over these lines is a rotation of 68°.

Use the figure shown, where .

31. Name three pairs of congruent angles.

32. If BC = 1.7 cm, which other side also measures 1.7 cm?

33. In the figure shown, .

a. Name all angles that are congruent to .

b. Name all angles that are supplementary to .

34. Braden is designing a special kite to use at night. He makes the drawing shown, where is the

perpendicular bisector of . He has 58 inches of glow-in-the-dark tape to put around the perimeter. If NG =

17 inches, what is the longest that IH can be?

____ 35. Given in the figure shown, what is the justification for the conclusion ?

A Definition of congruence

B Corresponding Parts of Congruent Figures (CPCF) Theorem

C Alternate Interior Angles Theorem

D Alternate Exterior Angle Theorem

36. Given the figure shown, tell whether the described construction is uniquely determined.

a. A circle with radius

b. A line through C parallel to

c. A bisector of

37. Refer to the figure shown. Find .

38. Given an equiangular hexagon, find the measure of an

a. interior angle.

b. exterior angle.

39. The figure shown has n-fold rotation symmetry.

a. n = _____.

b. What is the smallest magnitude of rotation that will map this figure onto itself?

Completely describe the reflection and rotation symmetry of each object, if any. If there is reflection

symmetry, draw the lines of symmetry.

40. Daisy:

41. In , an exterior angle to has measure 118 and an exterior angle to has measure 122. Which

side of is the longest?

42. In the figure shown, is a diameter and . Find

a. .

b. .

43. Tell whether each statement is always, sometimes but not always, or never true.

a. An isosceles triangle has three lines of symmetry.

b. A square is also a rhombus.

c. A trapezoid has exactly one pair of congruent base angles.

44. Suppose a regular polygon has n sides, where n is an odd number.

a. How many symmetry lines does the polygon have?

b. How many of those symmetry lines are angle bisectors?

c. How do your answers to Parts a and b change if n is an even number instead of an odd number?

45. Draw a hierarchy that relates the following: square, kite, rhombus, parallelogram, rectangle.

46. Find the measure of each angle of a regular decagon (10 sides).

47. Determine whether the frieze pattern shown below has each type of symmetry.

a. 180° rotation symmetry

b. reflection symmetry over a horizontal line

c. reflection symmetry over a vertical line

d. glide reflection symmetry

Determine whether the given triangles are congruent. If they are, write a congruence statement and

justify your answer with a triangle congruence theorem.

48.

49.

50.

51.

____ 52. Given with and cm, which of the following lengths for side does not

guarantee a unique triangle?

A YX = 7 cm B YX = 14 cm

C YX = 18 cm

D All of the above guarantee a unique triangle.

53. In parallelogram QUAD, , , , cm, and cm.

Find

a. .

b. .

c. QR.

54. Give the most specific name for each quadrilateral below.

a.

b.

55. a. Draw triangle PQR with , cm, and cm

b. If everyone in the class draws correctly, will all of the triangles be congruent? Explain.

Refer to the figures below. An apothem of the regular hexagon is shown.

a. b. c.

56. Find the perimeter of figures a, b, and c.

57. Find the area of figures a, b, and c.

58. Find the missing length in each triangle.

59. Give a value for x that makes the triangle

a. a right triangle.

b. an acute triangle.

60. Rectangle RECT is inscribed in as shown. Find the area of the circle.

61. The landscapers at the park used 114 bricks to make a circle around the edge of a flower garden.

a. If each brick is eight inches long, what is the circumference of the garden?

b. What is the radius of the garden to the nearest foot?

62. Draw right, front, and top views of the cubes shown.

a. right

b. front

c. top

Refer to the regular triangular prism shown.

63. Identify the following parts of the prism:

a. a base

b. a lateral face

c. a lateral edge

d. the height

64. If the perimeter of the base is 210 cm and SV = 105 cm, find the total surface area of the prism to the nearest

square centimeter.

65. Two different 3-dimensional figures each have eight faces.

a. One of the figures is a pyramid. How many vertices does it have?

b. The other figure is a prism. What shape is each base?

66. Given that planes P and Q intersect, what conclusion, if any, can be made about their intersection based on

the Point-Line-Plane Postulate?

67. Bryanna has a cylindrical tin container with a height of 4 inches and a diameter of 10 inches. She wants to

cover the outside of the tin on the bottom and sides with adhesive paper. Find the surface area of the contact

paper she will need. Round your answer to the nearest square inch.

68. Janna is considering transferring her flower from a tall skinny cylindrical vase to a shorter cylindrical vase,

and she wants to save the water because it has the flower food from the florist in it. If the first vase has a radius of 1 inch and the water fills it to a height of 10 inches, how high will the water reach in a vase with a

radius of 2 inches?

69. Arnie made a snowball with a diameter of 24 cm. Roberto made a snowball with a diameter of 18 cm. The

volume of Arnie’s snowball is how many times the volume of Roberto’s snowball? Round to the nearest

tenth.

70. An ice cream shop owner is trying to regulate how much ice cream his employees serve. For a small cone, the

owner instructs them to completely fill the cone and place half of a scoop on top of that. The scoop is a

sphere. The small cone and the scoop both have a diameter of 2 inches. The cone has a height of 5 inches. What volume of ice cream should be served?

71. Determine whether Cavalieri’s Principle can be used to equate the volume of a right hexagonal prism to each

of the following.

a. A right pentagonal prism with the same base area and height.

b. An oblique hexagonal prism with the same base area and height.

c. An oblique cylinder with the same base area and height.

d. A right hexagonal pyramid with the same base area and twice the height.

72. Consider the sphere shown. Find its

a. exact volume.

b. exact surface area.

Use the points A = (12, –5) and B = (–4, 3).

73. Find the distance AB.

74. Find the midpoint of .

75. Let K = (0, a), I = (2b, 0), T = (0, –a) and E = (–b, 0). Show that KITE is a kite.

76. The equation of a circle is .

a. What is the center of the circle?

b. What is the radius of the circle?

The points (3, 4, 7) and (7, 0, 9) are endpoints of a diameter of a sphere.

77. a. Find the center of the sphere.

b. Find the radius of the sphere.

78. Write an equation for the sphere.

79. In the figure shown, CG = 10. Find

a. AI.

b. BH.

80. In the figure shown, K, M, and O are the midpoints of their respective sides.

a. Explain why .

b. If KM = 12, find ON.

c. Name an angle that must be congruent to .

81. In the figure shown, . Find as many side lengths and angle measures as you can.

82. Suppose is a segment with slope 4 and length 6. is the image of under a size transformation of

magnitude k.

a. Find the slope of .

b. Find the length of .

____ 83. Which of the following proportions is not equivalent to ?

A

B

C

D

84. Consider the figure shown. If it is given that AD = 2.1, DB = 3.9, and AC = 10, find EC.

85. In the figure shown, bisects . Find IH.

86. Evaluate each trigonometric function. Give your answer as a numerical fraction.

a. sin(E)

b. cos(E)

c. tan(G)

87. Find the exact value of cos(30°).

88. Find BC to the nearest tenth.

89. A ramp makes a 15° angle with the ground. If the ramp covers a horizontal distance of 12 feet, how high is

the top of the ramp? Round your answer to the nearest inch.

90. Consider the figure shown. Given that JKLM is a rectangle, find

a. KN.

b. NM.

c. JN.

Geometry Final Exam - Study Guide

Answer Section

1. ANS:

false

PTS: 1

2. ANS:

0.7

PTS: 1

3. ANS:

a. 3

b. no

PTS: 1

4. ANS:

Yes; R is between Q and S because QR + RS = QS.

PTS: 1

5. ANS:

a. no

b. yes

c. yes

PTS: 1

6. ANS: A PTS: 1

7. ANS:

a. I

b. II

PTS: 1

8. ANS:

The figure is not a polygon because not every segment intersects two others at their endpoints. Two of the

segments intersect only one other segment.

PTS: 1

9. ANS:

a. Answers vary. Sample: x = 27

b. False; x = 6 is a counterexample.

PTS: 1

10. ANS:

a. If a number is divisible by 9, then it is divisible by 3.

b. true

PTS: 1

11. ANS:

a. false

b. Find one person with red hair who is not good at math.

PTS: 1

12. ANS:

PTS: 1

13. ANS:

a. Answers vary. Sample: and

b. Answers vary. Sample:

c. Answers vary. Sample:

d. Answers vary. Sample:

e. Answers vary. Sample:

PTS: 1

14. ANS: 123

PTS: 1

15. ANS: 54

PTS: 1

16. ANS:

a. 117

b. 297

c. 63

PTS: 1

17. ANS:

PTS: 1

18. ANS:

Two Perpendiculars Theorem: If two coplanar lines l and m are perpendicular to the same line, then they are parallel to each other.

PTS: 1

19. ANS:

a.

b.

c.

PTS: 1

20. ANS:

PTS: 1

21. ANS:

a. false

b. true

c. true

d. false

PTS: 1

22. ANS:

PTS: 1

23. ANS:

PTS: 1

24. ANS: Answers vary. Sample:

PTS: 1

25. ANS: True; reflections preserve angle measure.

PTS: 1

26. ANS: True; this is from the definition of reflection.

PTS: 1

27. ANS:

False. Answers vary. Sample:

PTS: 1

28. ANS:

Student may list any three of the following: angle measure, betweenness, collinearity, distance, orientation.

PTS: 1

29. ANS:

reflection or glide reflection

PTS: 1

30. ANS:

Answers vary. Sample:

PTS: 1

31. ANS:

any three of these:

PTS: 1

32. ANS:

PTS: 1

33. ANS:

a.

b.

PTS: 1

34. ANS:

12 in.

PTS: 1

35. ANS: C PTS: 1

36. ANS: a. no

b. yes

c. no

PTS: 1

37. ANS: 38

PTS: 1

38. ANS:

a. 120

b. 60

PTS: 1

39. ANS: a. 9

b. 40°

PTS: 1

40. ANS: four lines of symmetry; 4-fold (90°) rotation symmetry

PTS: 1

41. ANS:

PTS: 1

42. ANS:

a. 90

b. 34

PTS: 1

43. ANS: a. sometimes but not always

b. always

c. never

PTS: 1

44. ANS:

a. n

b. all of them (n)

c. There would still be n symmetry lines, but only half of them would be angle bisectors.

PTS: 1

45. ANS:

PTS: 1

46. ANS:

144

PTS: 1

47. ANS:

a. no

b. yes

c. no

d. yes

PTS: 1

48. ANS:

no

PTS: 1

49. ANS:

Yes; by HL or SsA.

PTS: 1

50. ANS:

Yes; by SAS.

PTS: 1

51. ANS: no

PTS: 1

52. ANS: C PTS: 1

53. ANS:

a. 45

b. 13

c. 18.5 cm

PTS: 1

54. ANS:

a. rhombus

b. isosceles trapezoid

PTS: 1

55. ANS:

a.

b. Yes; they are all congruent by SAS.

PTS: 1

56. ANS:

a. 40

b.

c. 84

PTS: 1

57. ANS:

a. 66

b. 225

c.

PTS: 1

58. ANS:

a.

b.

PTS: 1

59. ANS:

a. or

b. Answers vary. Any x such that will work.

PTS: 1

60. ANS:

225

PTS: 1

61. ANS:

a. 912 in. or 76 ft

b. 12 ft

PTS: 1

62. ANS:

a.

b.

c.

PTS: 1

63. ANS:

a. or

b. Answers vary. Sample: RUVS

c. Answers vary. Sample:

d. Answers vary. Sample: RU

PTS: 1

64. ANS:

26,294 cm2

PTS: 1

65. ANS:

a. 8

b. a hexagon

PTS: 1

66. ANS:

The intersection of planes P and Q is a line.

PTS: 1

67. ANS:

204 in2

PTS: 1

68. ANS:

2.5 in.

PTS: 1

69. ANS:

2.4

PTS: 1

70. ANS:

PTS: 1

71. ANS:

a. yes

b. yes

c. yes

d. no

PTS: 1

72. ANS:

a. 288 in3

b. 144 in2

PTS: 1

73. ANS:

PTS: 1

74. ANS:

(4, –1)

PTS: 1

75. ANS: Answers vary. Sample: KITE is a kite because it has two pairs of congruent sides.

PTS: 1

76. ANS: a. (4, –3)

b.

PTS: 1

77. ANS: a. (5, 2, 8)

b. 3

PTS: 1

78. ANS:

PTS: 1

79. ANS: a. 20

b. 15

PTS: 1

80. ANS:

a. by the Midsegment of a Triangle Theorem. Then, because they are

corresponding angles formed by parallel lines cut by a transversal.

b. 12

c. Answers vary. Sample:

PTS: 1

81. ANS:

, , , ,

PTS: 1

82. ANS:

a. 4

b. 6k

PTS: 1

83. ANS: D PTS: 1

84. ANS:

6.5

PTS: 1

85. ANS:

27.2

PTS: 1

86. ANS:

a.

b.

c.

PTS: 1

87. ANS:

PTS: 1

88. ANS:

14.8

PTS: 1

89. ANS:

39 in.

PTS: 1

90. ANS:

a. 7.2

b. 12.8

c. 9.6

PTS: 1