Geometry Quarter 4 Test Study Guide

1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement:

All right angles are congruent.

2. Find the measures of angles A, B, and C.

3. The measure of an angle is 64o. What is the measure of its complement? What is the measure of its supplement?

4. Write an equation of the line that passes through point P (2, -3) and is perpendicular to the line x – y = 4.

5. Using the diagram, give the coordinates of M if it is a midpoint.

6. A board 24 inches long is cut into two pieces in the ratio Find the length of each piece.

7. Given: ABC is isosceles with base ,AC BD bisects B

Prove: ABD CBD

8. Name five Theorems or Postulates that you can use to prove that two triangles are congruent.

9. A spotlight is mounted on a wall 7.4 feet above a security desk in an office building. It is used to light an

entrance door 9.3 feet from the desk. To the nearest degree, what is the angle of depression from the spotlight to

the entrance door?

10. Line l passes through the points (–3, 1) and (2, 5). If j l and k j, what is the slope of k? Explain your

reasoning.

11. Identify the property that makes the statement true. If MP = PQ and PQ = QR, then MP = QR.

12. For each set of numbers, determine whether the numbers represent the lengths of the sides of an acute triangle,

a right triangle, an obtuse triangle, or no triangle.

a) b) 26, 28, 51 c) 18, 38.5, 42.5

13. Find the values of x and y. 14. Find the measure of exterior angle

15. Find a, b, and h.

16. , and and bisect each other. Which triangle congruence theorem

or postulate could you use to prove that HML KMJ? Explain.

17. Tong is making a triangular shaped frame out 18. Determine whether the figures are similar.

of three strips of wood. One of the strips is 10

centimeters long and the second one is 15 centimeters

long. What are the possible lengths of the third

side?

19. Name a ray from Q through P.

20. A photo needs to be enlarged from an original with a length of 9 inches and a width of 7 inches to a size where

the new width is 14 inches. What is the new length? What is the scale factor?

21. Given: bisects RST . Find QR if and 22. Find the value of x.

(not drawn to scale)

23. Explain the difference between inductive and deductive reasoning.

24. In the diagram, are midsegments of triangle ABC.

Find the values of the variables if .

25. Line l is the perpendicular bisector of .

Find m M.

26. The length of one ramp is 16 feet. The vertical rise is 14 inches. Estimate the ramp’s horizontal distance and its

ramp angle.

27. Draw a Venn diagram showing the relationship between squares, rectangles, rhombuses, parallelograms, and

quadrilaterals.

28. Solve for x, given that . Is equilateral?

29. 1 and 2 are complementary, and 2 and 3 form a linear pair. If m 1 = , what is m 3? Explain your

reasoning.

30. Given the following statements, can you conclude that Becky plays basketball on Wednesday night?

(1) If it is Wednesday night, Becky goes to the gym.

(2) If Becky goes to the gym, she plays basketball.

31. Would HL, ASA, SAS, AAS, or SSS be used to justify that

the pair of triangles is congruent?

32. Which lines, if any, can be proved parallel given the following diagram?

For each conclusion, provide the justification.

33. Can the measurements 9.7 meters, 1.1 meters, and 6.9 meters be the lengths of the sides of a triangle?

34. Given the following, determine whether quadrilateral XYZW must be

a parallelogram. Justify your answer. .

35. a. Is the statement "If a quadrilateral is a rectangle, then it is a parallelogram" True or False?

b. Write the inverse of the statement in part (a) and tell if it is True or False.

36. Draw and label the angles and the sides of the two special triangles: 45o-45

o-90

o and 30

o-60

o-90

o.

37. True or False: The median and altitude of a triangle can never be the same line segment.

38. The altitude of an equilateral triangle is 6. What is the length of each side? Find the area of the triangle.

39. According to the Parallel Postulate, if there is a line and a point not on the line, then how many parallels to the

given line can be drawn through the point?

40. Wires are used to stabilize a telephone pole that is 50 feet high. A wire from the top of the pole to the ground is

63 feet long. To the nearest tenth of a foot, how far from the bottom of the pole is the wire anchored in the

ground?

41. Each interior angle of a regular n-gon has a measure of 156o. Find the value of n.

42. If is an altitude of PQR, what type of triangle is PQR?

43. Find the geometric mean of 8 and 12.

44. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 2340°.

45. Solve the right triangle: and find , b, and c.

46. Given that PQR ~ PST, explain why .

47. Given that a || b, what is the value of x? 48. JKL is an equilateral triangle.

(The figure may not be drawn to scale.) What is the length of KM ?

49. What is the measure of the smallest 50. Find the value of x.

interior angle in the quadrilateral?

51. A triangle has the given vertices. Classify the triangle by its sides. Then determine if it is a right triangle.

52. List all of the important characteristics of each quadrilateral.

a. square

b. rectangle

c. parallelogram

d. rhombus

e. trapezoid

f. kite

53. Consider an octagonal stop sign.

a. Find the sum of the interior angles of a stop sign.

b. Find the measure of one of the interior angles of a stop sign.

c. Find the measure of an exterior angle of a stop sign.

54. Which lines, if any, can be proved parallel given the following diagram?

55. Point S is between points R and T. P is the midpoint of . RT = 20 and PS = 4. Draw a sketch to show the

relationship between the specified segments. Find ST.

56. Given: is the perpendicular bisector of . Name three things that

you can conclude.

57. m SQR = ( )° and m PQR = ( )° and m SQP = 70°. 58. Solve the right triangle.

Find m SQR and m PQR.

59. List each type of quadrilateral for which the statement is always true:

The diagonals are congruent.

60. In ,RSTU RS is 3 centimeters shorter than ST . The Perimeter of RSTU is 42 centimeters.

Find RS and ST .

61. In the diagram, °, °, , and .

Is there enough information given to show that quadrilateral ABCD

is an isosceles trapezoid? Explain.

62. If p q, solve for x.

63. If one angle of a triangle is larger than another angle, then the side opposite the larger

angle is longer than the side opposite the smaller angle. Use this fact to help you list the

sides of triangle STU in order from least to greatest. (The figure may not be drawn to scale.)

64. Two triangles are similar. The height of the smaller triangle is 4 units and the height of the larger triangle is 6

units. If the area of the smaller triangle is 24 square units, what is the area of the larger triangle?

65. Let p be “it is raining”, let q be “it is thundering”, and let r be “we cannot swim”. What is p q ?

66. form a linear pair. °. Find .

67. Find in the diagram, if ° and °.

68. Find the area of the isosceles triangle with side lengths 17 meters, 17 meters, and 30 meters.

69. Calculate the slope of the line. Does it matter which points are used?

Why or why not?

70. A building casts a shadow 260 meters long. At the same time, a pole 3 meters high casts a shadow 15 meters

long. What is the height of the building?

71. Tell whether each pair of triangles is similar. Explain your reasoning.

72. In and In and State whether the

triangles are similar, and if so, write a similarity statement.

73. Identify the hypothesis and conclusion of the statement: If today is Friday, then yesterday was Thursday.

74. The ratios of the side lengths of triangle ABC are 7:9:12 (AB:AC:BC). Solve for x.

75. True or False: If a quadrilateral is a parallelogram, then opposite angles are complementary.

76. Write a logical conclusion from the following statements:

If it is raining, then we will watch a movie. If we watch a movie, then we will eat popcorn.

77. Find the appropriate symbol to place in the blank. (not drawn to scale)

AB __ AC

78. Find the side lengths of the kite. 79. Find the value of x.

80. State the third congruence statement that is needed to prove that the two triangles are congruent using the given

postulate or theorem.

81. Find the value of each variable.

a) b) c) d)

82. Find RS in C. Explain your reasoning. 83. is tangent to O at A (not drawn to scale).

Find the length of the radius r, to the nearest tenth.

84. Find the .m G 85. Find the values of the variables.

86. Find the value of x.

a) b) c)

87. Write the standard equation of a circle with a center (3, -2) and a point on the circle (23, 19).

88. In the diagram, is a radius of circle R. Is tangent to circle R? Explain.

89. Graph the equation: 2 2( 5) ( 3) 9x y 90. Find the value of x.

91. Find the value of x.

a) b) c)

92. The rule for this transformation 93. What are the coordinates of the vertices

of onto is _________ when the figure is reflected in line m?

94. The translation vector is = . If the image of A is find the coordinates of point A.

95. The vertices of are .

Reflect the triangle in the line .

Then find the image of ABC after a dilation with its center at the origin

and a scale factor of 2.

96. A triangle has vertices . 97. The vertices of are (2,5),B(6,5)A

Find the coordinates of the vertices of the images of and (3,8)C . Find the image of ABC

after rotations of 90°, 180° and 270° about the origin. after the glide reflection:

Translation:

Reflection: in

98. Which transformation(s) are used in the 99. State whether the following figure has line

tessellation below? Which shape, 1, 2, 3, or 4, symmetry, rotational symmetry, both kinds

belongs in the location indicated of symmetry, or neither kind of symmetry.

by the arrow?

Geometry Quarter 4 Test Study Guide – Answer Key

1. If the angles are right angles, then they are congruent. If the angles are congruent, then they are right angles.

If the angles are not right angles, then they are not congruent. If the angles are not congruent, then they are not right angles.

2. m A = 103°, m B = 77°, m C = 46° 3. 26o, 116o 4. 1y x 5.

6. 7. 1) ABC is isosceles with base AC, BD bisects B (Given)

2) AB BC (Definition of isosceles triangle)

3) ABD CBD (Definition of angle bisector)

4) BD BD (Reflexive Property of Segment Congruence)

5) ( )ABD CBD SAS

8. SSS, SAS, HL, AAS, ASA 9. 39° 10. – . The slope of line l is = . Since j l, the slope of j is also . Since k

j, the slope of k is the negative reciprocal of , which is – . 11. Transitive Property of Equality

12. A. acute triangle, B. obtuse triangle, C. right triangle 13. x = 11, y = 14. 89°

15. a = 14, b = , h =

16. SAS Congruence Postulate. Since and bisect each other, and . because they

are vertical angles. Since you know that 2 pairs of sides and the included angle are congruent, you can use the SAS Congruence Postulate

to prove the triangles are congruent.

17. 5 < x < 25 18. The figures are not similar. 19.

20. new length = 18 inches; scale factor = 2 21. 34 22. 6

23. Inductive reasoning is the process of making a generalization based on a number of similar cases or specific patterns. Deductive

reasoning is the process of following a series of logical steps, often beginning with some given or known information, that leads to a specific

conclusion.

24. X=8, Y= 2, Z=15 25. 64° 26. 7.7, 61o

27. Diagrams vary. 28. x = 8; no

29. . Since 1 and 2 are complementary, the sum of their measures is . So, m 2 = . Since 2 and 3 form a linear pair,

they are supplementary and the sum of their measures is . So, m 3 = – = .

30. yes 31. AAS 32. , Consecutive Interior Angles Converse 33. No

34. Yes. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

35. a. True b. If a quadrilateral is not a rectangle, then it is not a parallelogram. False 36.

37. False 38. , 12 3 39. Exactly one 40. 38.3

41. 15n 42. A right triangle 43. 4 6

44. 15 45.

46. Since PST ~ PQR, PST Q and PTS R. Since the pairs of angles are corresponding angles, by the

Corresponding Angles Converse Postulate.

47. 71 48. 9 3 49. 40 50. 124°

51. Right scalene triangle

52. The following characteristics for each quadrilateral might be indicated.

a. four congruent sides, four right (congruent) angles, opposite sides parallel, congruent diagonals, diagonals are the perpendicular

bisectors of each other.

b. opposite sides congruent and parallel, four right (congruent) angles, congruent diagonals, diagonals bisect each other

c. opposite sides congruent and parallel, opposite angles congruent, diagonals bisect each other

d. four congruent sides, opposite sides parallel, opposite angles congruent, diagonals are the perpendicular bisectors of each other

e. one pair of opposite sides parallel

f. two pairs of adjacent sides congruent

53. a. 1080°, b. 135°, c. 45° 54. No lines can be proved parallel from the given information.

55. 12

56. Any three of the following: ; ;

= , =

57. m SQR = 20° and m PQR = 50° 58. 24.4 , 65.6 , 12.1A C AC

59. square, rectangle 60. 12, 9

61. Yes, enough information is given to show ABCD is an isosceles trapezoid. ABCD is a trapezoid because so

. and are not congruent so is not parallel to . By definition ABCD is a trapezoid. The diagonals of

trapezoid ABCD are congruent because . So, ABCD is an isosceles trapezoid by Theorem 8.16.

62. 12 63. 64. 54 square units 65. If it is raining, then it is not thundering.

66. 107° 67. ° 68.

69. ; no; the slope ratio is the same for any two points on a line. 70. 52 meters

71. Yes; The two right angles are congruent, and since parallel lines are given the alternate interior angles are congruent, so the triangles are

similar by the AA Similarity Postulate

72. not similar 73. hypothesis: today is Friday, conclusion: yesterday was Thursday 74. 6 75. False

76. If it is raining, we will eat popcorn.

77. 78. 5 5; 461XY YZ WX WZ 79. X= 2.3

80. , ;F J ,D G

81. a) 9.2 b) 10 c) x = 24, y = 24 2 d) 7 3 21

,2 2

x y