Name: ______________________ Class: _________________ Date: _________ ID: A

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Mr. Boushee's Exciting and Thrilling Gemoentry Final Exam Review of First Semester

Short Answer

1. Find the length of the midsegment. The diagram is not to scale.

2. Write the tangent ratios for ∠P and ∠Q.

3. Line r is parallel to line t. Find m∠5. The diagram is not to scale.

Name: ______________________ ID: A

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4. Find the value of x. The diagram is not to scale.

5. Find the circumference of the circle in terms of π.

6. Judging by appearance, name an acute angle, an obtuse angle, and a right angle.

7. m∠A = 8x − 2, m∠B = 2x − 8, and m∠C = 94 − 4x. List the sides of ΔABC in order from shortest to longest.

8. Write the ratios for sin A and cos A.

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Solve for x.

9.

10.

11.

12. Find a counterexample to show that the conjecture is false.Conjecture: Any number that is divisible by 2 is also divisible by 4.

13. Name an angle supplementary to ∠EOA.

14. Find the value of the variable if m Ä l, m∠1 = 3x + 30 and m∠5 = 6x + 45. The diagram is not to scale.

Name: ______________________ ID: A

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15. Supplementary angles are two angles whose measures have sum ____.Complementary angles are two angles whose measures have sum ____.

16. Based on the pattern, what is the next figure in the sequence?

17. Name the four labeled segments that are skew to EG.

18. For the following true conditional statement, write the converse. If the converse is also true, combine the statements as a biconditional.If x = 7, then x2 = 49.

19. What else must you know to prove the triangles congruent by ASA? By SAS?

20. Are the triangles similar? If so, explain why.

21. If possible, use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible.Statement 1: If x = 3, then 5x – 7 = 8.Statement 2: x = 3

Name: ______________________ ID: A

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22. The triangular playground has angles whose measures are in the ratio 9 : 3 : 8. What is the measure of the smallest angle?

23. Is there enough information to conclude that the two triangles are congruent? If so, what is a correct congruence statement?

24. Find the perimeter of ΔABC with vertices A(–8, –4), B(–4, –4), and C(–8, –1).

25. ∠PMN ≅ ?

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Find the value of x. Round to the nearest tenth.

26.

27.

28.

29. Find the values of x, y, and z. The diagram is not to scale.

30. Name the ray that is opposite BD→⎯⎯⎯

.

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31. The Sears Tower in Chicago is 1450 feet high. A model of the tower is 24 inches tall. What is the ratio of the height of the model to the height of the actual Sears Tower?

32. Use the Law of Detachment to draw a conclusion from the two given statements.

If two angles are supplementary, then the sum of their measures is 180°.

∠H and ∠G are supplementary.

33. From the information in the diagram, can you prove ΔFDG ≅ ΔFDB? Explain.

Solve for a and b.

34.

35.

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36. Given AE Ä BD, solve for x.The diagram is not drawn to scale.

Write the sum of the two vectors as an ordered pair.

37. −3〈 , 5〉 and −4〈 , −1〉

38. 5〈 , −2〉 and 0〈 , 0〉

39. −6〈 , 5〉 and 6〈 , −5〉

Solve the proportion.

40. 25

= m15

41. 6

a=

18

21

42. 3y − 8

12=

y

5

43. A triangle has sides of lengths 9, 7, and 12. Is it a right triangle? Explain.

44. Use the information in the diagram to determine the height of the tree to the nearest foot.

45. Find the distance between points P(8, 4) and Q(1, 1) to the nearest tenth.

46. M(8, 5) is the midpoint of RS. The coordinates of S are (9, 6). What are the coordinates of R?

Name: ______________________ ID: A

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47. Find the values of x and y.

48. If T is the midpoint of SU, find the values of x and ST. The diagram is not to scale.

49. Noam walks home from school by walking 8 blocks north and then 6 blocks east. How much shorter would his walk be if there were a direct path from the school to his house? Assume that the blocks are square.

50. Find x to the nearest tenth.

51. Find the value of x. The diagram is not to scale.

Name: ______________________ ID: A

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Find the value of x. Round the length to the nearest tenth.

52.

53.

54.

55. Name the plane represented by the top of the box.

56. Is the statement a good definition? If not, find a counterexample.A square is a figure with two pairs of parallel sides and four right angles.

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57. List the sides in order from shortest to longest. The diagram is not to scale.

58. BD bisects ∠ABC. m∠ABC = 9x. m∠ABD = 4x + 27. Find m∠DBC.

59. What is the measure of a base angle of an isosceles triangle if the vertex angle measures 46° and the two congruent sides each measure 21 units?

60. 6 and 9

61. The students in Mr. Collin’s class used a surveyor’s measuring device to find the angle from their location to the top of a building. They also measured their distance from the bottom of the building. The diagram shows the angle measure and the distance. To the nearest foot, find the height of the building.

Name: ______________________ ID: A

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62. Which point is the midpoint of AE?

63. The complement of an angle is 69°. What is the measure of the angle?

Fill in each missing reason.

64. Given: 9x − 7y = −2; x = −3

Prove: −257

= y

9x − 7y = −2; x = −3 a. ________

−27 − 7y = −2 b. ________

−7y = 25 c. ________

y = −257

d. ________

−257

= y e. ________

65. Given: m∠PQR = x + 3, m∠SQR = x + 1, and m∠PQS = 100.Find x.

m∠PQR + m∠SQR = m∠PQS a. _____x + 3 + x + 1 = 100 b. Substitution Property

2x + 4 = 100 c. Simplify2x = 96 d. _____x = 48 e. Division Property of Equality

Name: ______________________ ID: A

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66. What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate?

67. Given ΔABC ≅ ΔPQR, m∠B = 5v + 6, and m∠Q = 6v − 7, find m∠B and m∠Q.

68. Given: PQ Ä BC. Find the length of AQ. The diagram is not drawn to scale.

Describe the vector as an ordered pair. Give the coordinates to the nearest tenth. (Not drawn to scale)

69.

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Find the length of the missing side. The triangle is not drawn to scale.

70.

71. If m∠EOF = 22 and m∠FOG = 21, then what is the measure of ∠EOG? The diagram is not to scale.

72. If a

b=

3

5, then 5a = ____.

73. Michele wanted to measure the height of her school’s flagpole. She placed a mirror on the ground 48 feet from the flagpole, then walked backwards until she was able to see the top of the pole in the mirror. Her eyes were 5 feet above the ground and she was 12 feet from the mirror. Using similar triangles, find the height of the flagpole to the nearest tenth of a foot.

74. Identify the hypothesis and conclusion of this conditional statement:If today is Wednesday, then tomorrow is Thursday.

75. Use the information given in the diagram. Tell why MP ≅ NO and ∠NOM ≅ ∠PMO.

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76. Name the second largest of the four angles named in the figure (not drawn to scale) if the side included by ∠1 and ∠2 is 10 cm, the side included by ∠2 and ∠3 is 17 cm, and the side included by ∠3 and ∠1 is 13 cm.

Explain why the triangles are similar. Then find the value of x.

77.

78.

79. Find the lengths of the missing sides in the triangle. Write your answers as integers or as decimals rounded to the nearest tenth.

Name: ______________________ ID: A

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Use compass directions to describe the direction of the vector. (Not drawn to scale)

80.

81. Which plane is parallel to plane EFHG?

82. Use the information in the diagram to determine the height of the tree. The diagram is not to scale.

83. Find the area of a rectangle with base 6 yd and height 5 ft.

84. Alfred is practicing typing. The first time he tested himself, he could type 26 words per minute. After practicing for a week, he could type 32 words per minute. After two weeks he could type 38 words per minute. Based on this pattern, predict how fast Alfred will be able to type after 4 weeks of practice.

85. Patrick wants to put a fence around his rectangular garden. His garden measures 34 feet by 54 feet. The garden has a path around it that is 3 feet wide. How much fencing material does Patrick need to enclose the garden and path?

86. Given points A(2, 3) and B(–2, 5), explain how you could use the Distance Formula and an indirect argument to show that point C(0, 3) is NOT the midpoint of AB.

87. Given ΔQRS ≅ ΔTUV, QS = 2v + 4, and TV = 5v − 8, find the length of QS and TV.

Name: ______________________ ID: A

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88. Q is equidistant from the sides of ∠TSR. Find the value of x. The diagram is not to scale.

Find the value of x. Round your answer to the nearest tenth.

89.

90.

91. ΔQRS ∼ ΔTUV. What is the measure of ∠V?

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92. Find the value of x.

93. Name the Property of Congruence that justifies the statement:If ∠P ≅ ∠Q and ∠Q ≅ ∠R, then ∠P ≅ ∠R.

94. Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible.I will score well on the exam if I study for 1 hour each day.I will score well on the exam.

95. Find the length of AB, given that DB is a median of the triangle and AC = 56.

96. Name the Property of Equality that justifies the statement:If m = n, then n = m .

97. Find the center of the circle that you can circumscribe about ΔEFG with E(6, 4), F(6, 2), and G(8, 2).

98. What is the intersection of plane STUV and plane UYXT?

99. Name the Property of Congruence that justifies the statement:If YZ ≅ WX, then WX ≅ YZ.

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100. Determine whether the conditional and its converse are both true. If both are true, combine them as a biconditional. If either is false, give a counterexample.If an angle is a right angle, its measure is 90.If an angle measure is 90, the angle is a right angle.

101. DF bisects ∠EDG. Find the value of x. The diagram is not to scale.

102. State whether ΔABC and ΔAED are congruent. Justify your answer.

103. Find the values of x and y.

104. ∠1 and ∠2 are supplementary angles. m∠1 = x − 23, and m∠2 = x + 71. Find the measure of each angle.

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Find the length of the missing side. Leave your answer in simplest radical form.

105.

106. What is the negation of this statement?Miguel’s team won the game.

107. Write the two conditional statements that make up the following biconditional.I drink juice if and only if it is breakfast time.

108. If BCDE is congruent to OPQR, then DE is congruent to ? .

109. Classify ΔABC by its angles, when m∠A = 20, m∠B = 88, and m∠C = 72.

Find the value of x to the nearest degree.

110.

111. Chris is adding a ribbon border to the edge of his kite. Two sides of the kite measure 6.8 inches, while the other two sides measure 18.6 inches. How much ribbon does Chris need?

112. Miguel is driving his motorboat across a river. The speed of the boat in still water is 9 mi/h. The river flows directly south at 5 mi/h. If Miguel heads directly west, what are the boat’s resultant speed and direction? (Not drawn to scale)

113. Find the coordinates of the midpoint of the segment whose endpoints are H(10, 8) and K(6, 2).

114. Write the inverse of this statement:If a number is divisible by two, then it is even.

Name: ______________________ ID: A

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115. Write this statement as a conditional in if-then form:All triangles have three sides.

116. Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements.If an elephant weighs more than 2,000 pounds, then it weighs more than Jill’s car.If something weighs more than Jill’s car, then it is too heavy for the bridge.Smiley the Elephant weighs 2,150 pounds.

117. Two sides of a triangle have lengths 8 and 17. What must be true about the length of the third side?

118. ∠DFG and ∠JKL are complementary angles. m∠DFG = x + 7, and m∠JKL = x − 5. Find the measure of each angle.

119. What is the measure of the vertex angle of an isosceles triangle if one of its base angles measures 44°?

120. Find the value of x. The diagram is not to scale.

121. Find the area of the circle in terms of π.

122. To approach the runway, a small plane must begin a 10° descent starting from a height of 1624 feet above the ground. To the nearest tenth of a mile, how many miles from the runway is the airplane at the start of this approach?

State whether the triangles are similar. If so, write a similarity statement and the postulate or theorem you used.

123. In ΔQRS, QR = 16, RS = 64, and m∠R = 29. In ΔUVT, VT = 8, TU = 32, and m∠T = 29.

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124.

125. What additional information will allow you to prove the triangles congruent by the HL Theorem?

126. Are points D, H, and C collinear or noncollinear?

127. If EF = 4x − 20, FG = 3x − 8, and EG = 28, find the values of x, EF, and FG. The drawing is not to scale.

128. Based on the pattern, what are the next two terms of the sequence?2, 8, 14, 20, . . .

Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form.

129.

Not drawn to scale

Name: ______________________ ID: A

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130. Find the value of x and y rounded to the nearest tenth.

131. Identify parallel segments in the diagram.

132. If the perimeter of a square is 96 inches, what is its area?

133. Complete the indirect proof.

Given: Bobby and Kina together hit at least 30 home runs. Bobby hit 18 home runs.Prove: Kina hit at least 12 home runs.

Assume Kina hit a.___ than 12 home runs. This means Bobby and Kina combined to hit at most b.____ home runs. This contradicts the given information that c. _____. The assumption is false. Therefore, Kina d. ______.

134. A model is made of a car. The car is 2 meters long and the model is 3 centimeters long. What is the ratio of the length of the car to the length of the model?

135. In ΔABC, G is the centroid and BE = 12. Find BG and GE.

Name: ______________________ ID: A

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Find the value of x. Round to the nearest degree.

136.

137. A triangle has side lengths of 8 cm, 15 cm, and 24 cm. Classify it as acute, obtuse, or right.

138. What is the contrapositive of this statement?If you have sea water, you can make salt.

Find the geometric mean of the pair of numbers.

139. 289 and 4

140. What is the converse of the following conditional?If a point is in the fourth quadrant, then its coordinates are negative.

ID: A

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Mr. Boushee's Exciting and Thrilling Gemoentry Final Exam Review of First SemesterAnswer Section

SHORT ANSWER

1. 64

2. tan P = 512

; tan Q = 125

3. 141 4. 66 5. 96π in. 6. ∠V, ∠U, ∠W

7. AC; AB; BC

8. sin A = 4850

, cos A = 1450

9. 70 10. 2 35 11. 8 12. 18 13. ∠AOC 14. –5 15. 180; 90 16.

17. BF, DH, CD, AB 18. If x2 = 49, then x = 7. False

19. ∠ACD ≅ ∠CAB; AD ≅ BC 20. yes, by AA 21. 5x – 7 = 8 22. 27 23. Yes; ΔACB ≅ ΔADC. 24. 12 units 25. ∠CAB 26. 11.9 27. 17 28. 33.1 29. x = 83, y = 71, z = 97

30. BA→⎯⎯

31. 1 : 725 32. m∠H + m∠G = 180 33. yes, by ASA

ID: A

2

34. a =400

21, b =

580

21

35. a =9

2, b =

15

2

36. 76

7 37. −7〈 , 4〉 38. 5〈 , −2〉 39. 0〈 , 0〉 40. 6 41. 7

42. 40

3 43. no; 9 2 + 7 2 ≠ 12 2

44. 80 ft 45. 7.6 46. (7, 4) 47. x = 90, y = 26 48. x = 10, ST = 80 49. 4 blocks 50. 28.8 51. x = 24 52. 692.8 m 53. 6.7 ft 54. 9.7 m 55. ACD 56. No; a rectangle is a counterexample. 57. LJ, JK, LK 58. 243 59. 67° 60. 3 6 61. 308 ft 62. C 63. 21° 64. a. Given

b. Substitution Propertyc. Addition Property of Equalityd. Division Property of Equalitye. Symmetric Property of Equality

65. Angle Addition Postulate; Subtraction Property of Equality

66. AC ⊥ BD 67. 71 68. 6

ID: A

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69. 3.9〈 , −7〉 70. 8 71. 43 72. 3b 73. 20 ft 74. Hypothesis: Today is Wednesday. Conclusion: Tomorrow is Thursday. 75. Given, Given 76. ∠1

77. AA Postulate; 11 12

78. AA Postulate; 14 25

79. x = 5.7, y = 4 80. 14° east of north 81. plane ABDC 82. 90 ft 83. 90 ft 2

84. 50 words per minute 85. 200 ft 86. Assume that C(0, 3) is the midpoint of AB. By the Distance Formula,

AC = (2 − 0) 2 + (3 − 3) 2 = 2 and BC = (−2 − 0) 2 + (5 − 3) 2 = 8 .AC ≠ BC which contradicts the assumption that C is the midpoint of AB.Therefore, C is not the midpoint of AB.

87. 12 88. 2 89. 11.9 90. 22 91. 70° 92. –19 93. Transitive Property 94. not possible 95. 28 96. Symmetric Property 97. (7, 3)

98. TX→←⎯⎯

99. Symmetric Property 100. Both statements are true. An angle is a right angle if and only if its measure is 90. 101. 15 102. yes, by either SSS or SAS 103. x = 18, y = 26 104. ∠1 = 43, ∠2 = 137

105. 3 29 cm 106. Miguel’s team did not win the game.

ID: A

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107. If I drink juice, then it is breakfast time.If it is breakfast time, then I drink juice.

108. QR 109. acute 110. 77 111. 50.8 in. 112. 10.3 mi/h; 29° south of west 113. (8, 5) 114. If a number is not divisible by two¸ then it is not even. 115. If a figure is a triangle, then it has three sides. 116. Smiley is too heavy for the bridge. 117. less than 25 118. ∠DFG = 51, ∠JKL = 39 119. 92° 120. 79 121. 576π in.2 122. 1.8 mi 123. The triangles are not similar. 124. ΔABC ∼ ΔMNO; SSS 125. AC ≅ DC 126. noncollinear 127. x = 8, EF = 12, FG = 16 128. 26, 32 129. x = 39, y = 13 3 130. x = 24.0, y = 46.4 131. BD Ä AE, DF Ä AC, BF Ä CE,

132. 576 in. 2

133. a. fewerb. 29c. Bobby and Kina together hit at least 30 home runsd. hit at least 12 home runs

134. 200 : 3 135. BG = 4, GE = 8 136. 28 137. obtuse 138. If you can’t make salt, you do not have sea water. 139. 34 140. If the coordinates of a point are negative, then the point is in the fourth quadrant.