Name: Class: Date: ID: A Geometry First Semester Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find m 1 in the figure below. PQ parallel. and RS are 4. If m AOC = 48 and m BOC = 21, then what is the measure of AOB? a. 121 b. 31 c. 59 d. 131 2. Which is the appropriate symbol to place in the blank? (not drawn to scale) AB AO a. 24 b. 32 c. 27 d. 29 5. ΔABD ΔCBD. Name the theorem or postulate that justifies the congruence. a. < b. > c. = d. not enough information 3. Find the value of x: a. HL b. SAS c. AAS d. ASA 6. ΔABD ΔCBD. Name the theorem or postulate that justifies the congruence. a. 88 b. 145 c. 35 d. 127 a. AAS b. SAS c. HL d. ASA 1
7. Find the value of y that will allow you to prove that CD below is parallel to EF if the measure of 1 is Ê ËÁ 4y 12 ˆ and the measure of 2 is 48. (The figure may not be drawn to scale.) 10. Which postulate or theorem can be used to determine the length of RT? a. 22 b. 69 c. 23 d. 36 8. In the diagram below, KF is the perpendicular bisector of GH. Then KGF. a. ASA Congruence Postulate b. AAS Congruence Theorem c. SSS Congruence Postulate d. SAS Congruence Postulate 11. Find the value of x: a. FKG b. KF c. KHF d. KFH 9. The distance between points A and B is. a. 92 b. 144 c. 128 d. 36 12. Find the values of x and y. a. x = 16, y = 98 b. x = 16, y = 82 c. x = 82, y = 98 d. x = 82, y = 62 a. 13 b. 11 c. 85 d. 85 2
13. What is the measure of each base angle of an isosceles triangle if its vertex angle measures 44 degrees and its 2 congruent sides measure 18 units? Refer to the figure below for the next two questions. Given: AF FC, ABE EBC a. 68 b. 46 c. 136 d. 44 14. If KF is the altitude of GKH and GK HK, then GKF. a. KHF b. FGK c. KF d. HKF 15. Name an angle complentary to COD. 16. An altitude of ΔGCF is. a. CF b. FG c. CD d. GF 17. A perpendicular bisector of ΔABC is. a. BE b. BF c. BD d. GF 18. In the figure, 6 and 2 are. a. COB b. DOB c. DOE or AOC d. DOC or AOE a. alternate interior angles b. consecutive interior angles c. alternate exterior angles d. corresponding angles 19. Two sides of a triangle have sides 4 and 13. The length of the third side must be greater than and less than. a. 4, 13 b. 9, 17 c. 8, 18 d. 3, 14 3
20. If AB = 14 and AC = 27, find the length of BC. 25. Refer to the figure. a. 14 b. 41 c. 13 d. 3 21. Which best describes the relationship between Line 1 and Line 2? Line 1 passes through Ê Ë Á 2, 7 ˆ and Ê ËÁ 4, 3 ˆ Line 2 passes through Ê Ë Á 4,9 ˆ and Ê ËÁ 2,13 ˆ a. perpendicular b. They are the same line. c. parallel d. neither perpendicular nor parallel 22. Which step in an indirect proof (proof by contradiction) is, point out the assumption must be false and therefore the conclusion must be true? a. first b. second c. third d. none of the above 23. If m GOH = 24 and m FOH = 50, then what is the measure of FOG? The longest segment is. a. MP b. NM c. ML d. LN 26. Let B be between C and D. Use the Segment Addition Postulate to solve for w. CB = 4w 4 BD = 2w 8 CD = 24 a. w = 6 b. w = 10 c. w = 2 d. w = 4 27. In the figure, 8 and 2 are. a. 31 b. 28 c. 23 d. 26 24. 1 and 2 are supplementary angles. 1 and 3 are vertical angles. If m 2 = 72, what is m 3? a. alternate exterior angles b. consecutive interior angles c. corresponding angles d. alternate interior angles a. 18 b. 72 c. 108 d. 28 4
28. What is the measure of each base angle of an isosceles triangle if its vertex angle measures 42 degrees and its 2 congruent sides measure 17 units? 31. Solve for x. a. 138 b. 69 c. 42 d. 48 29. In the figure shown, m CED = 63. Which of the following statements is false? a. 3 b. 6 c. 1 d. 2 32. Use the figure below to solve for x. G = x O = (2x + 21) a. 23 b. 33.5 c. 53 d. 67 a. AED and BEC are vertical angles. b. m BEC = 107 c. BEC and AEB are adjacent angles. d. m AEB = 63 30. If RS = 51 and QS = 87.3, find QR. a. 36.3 b. 26.3 c. 51 d. 138.3 33. A line L 1 has slope 4. The line that passes 9 through which of the following pairs of points is parallel to L 1? a. (6, 3) and (2, 6) b. (12, 1) and (2, 8) c. ( 5, 2) and (6, 6) d. ( 3, 2) and (6, 6) 34. Given: AE bisects DAB. Find ED if CB = 12 and CE = 16. (not drawn to scale) a. 28 b. 192 c. 20 d. 4 5
35. Let E be between F and G. Use the Segment Addition Postulate to solve for u. FE = 7u 6 EG = 2u 21 FG = 36 a. u = 4 b. u = 7 c. u = 11 d. u = 3 36. Which best describes the relationship between the line that passes through (4, 4) and (7, 6) and the line that passes through ( 3, 6) and (0, 4)? a. parallel b. neither perpendicular nor parallel c. same line d. perpendicular 37. In the figure, l Ä n and r is a transversal. Which of the following is not necessarily true? 38. How many triangles are formed by drawing diagonals from one vertex in the figure? Find the sum of the measures of the angles in the figure. a. 8 2 b. 2 6 c. 5 3 d. 7 4 a. 7, 1260 b. 6, 1260 c. 6, 1080 d. 7, 1080 39. The measure of each exterior angle of a regular octagon is. a. 22.5 b. 67.5 c. 45 d. 135 6
40. For parallelogram PQLM below, if m PML = 83, then m PQL =. 41. Find the value of the variables in the parallelogram. a. m PQM b. 83 c. 97 d. m QLM a. x = 54, y = 15.5, z = 149 b. x = 41, y = 31, z = 108 c. x = 15.5, y = 54, z = 149 d. x = 31, y = 41, z = 108 Short Answer 42. Find the slope of a line perpendicular to the line containing the points (12, -8) and (5, 4). 43. 1 and 2 form a linear pair. m 1=36. Find m 2. 44. Find the coordinates of the midpoint of the segment with the given pair of endpoints. J(6, 6); K(2, 4) 45. Write the contrapositive of the following statement. If a number is not divisible by two, then it is not even. 46. m RPQ = (2x + 7) and m OPQ = (7x 3) and m RPO = 67. Find m RPQ and m OPQ. 48. Two sides of a triangle have lengths 14 and 10. What are the possible lengths of the third side x? 49. Two sides of a triangle have lengths 28 and 67. Between what two numbers must the measure of the third side fall? 50. If m AOB = 27 and m AOC = 49, then what is the measure of BOC? 51. Find the value of x. 47. Line l is the perpendicular bisector of MN. Find m M. 52. 1 and 2 are supplementary angles. 1 and 3 are vertical angles. m 2=67. Find m 3. 7
53. Decide if the argument is valid or invalid. If the argument is valid, tell which rule of logic is used. If the argument is invalid, tell why. 1) If a figure is a quadrilateral, then it is a polygon. 2) I have drawn a figure that is a polygon. 3) Therefore, the figure I drew is a quadrilateral. 54. From the given true statements, make a valid conclusion using either the Law of Detachment or the Law of Syllogism: 1) If Ahmed can get time off work, he will go to Belize. 2) If Ahmed goes to Belize, Jake will go with him. Ahmed will get time off work. 55. Decide if the argument is valid or invalid. If the argument is valid, tell which rule of logic is used. If the argument is invalid, tell why. 1) If today is a holiday, then we do not have school. 2) Today is not a holiday. 3) Therefore, we do have school. 56. Write the inverse of the following statement. If a number is not even, then it is not divisible by two. 57. Decide whether Line 1 and Line 2 are parallel, perpendicular, or neither. Line 1 passes through (1, 2) and ( 3, 4) Line 2 passes through ( 3, 7) and ( 1, 3) 58. Identify the hypothesis and conclusion of the statement. If tomorrow is Friday, then today is Thursday. 59. m QOP = (2x + 6) and m NOP = (9x 1) and m QON = 60. Find m QOP and m NOP. 63. Tell whether the converse of the following statement is True or False. If it is false, give a counterexample. "A number is divisible by 2 if the number is divisible by 4." 64. Find the measures of all three angles of the triangle. 60. Use the Law of Detachment to write the conclusion to be drawn from the two pieces of given information: 1) If you drive safely, then the life you save may be your own. 2) Shani drives safely. 65. Solve for x. 61. Solve for x: 66. If QR is an altitude of ΔPQR, what type of triangle is ΔPQR? 62. 1 and 2 are complementary, and 2 and 3 form a linear pair. If m 3 = 140, what is m 1? Explain your reasoning. 8
67. Line l is the perpendicular bisector of MN. Find the value of x. 68. Find AB and BC in the situation shown below. AB = x + 16, BC = 5x + 10, AC = 56 69. Find the midpoint of the segment with endpoints (1, 1) and ( 15, 17). 70. Given the following statements, can you conclude that Marvin listens to the radio on Monday night? If so, state which law justifies the conclusion. If not, write no conclusion. (1) If it is Monday night, Marvin stays at home. (2) If Marvin stays at home, he listens to the radio. 71. Identify the hypothesis and conclusion of the statement. If today is Tuesday, then yesterday was Monday. 72. Find the length of AB. 75. Identify the longest side of ΔABC. 76. QS bisects RQT m RQS = (4x 2) and m SQT = (x + 13). 73. Using the diagram, give the coordinates of M if it is a midpoint. 74. The measure of an angle is 32 less than the measure of its supplement. Find the measure of the angle and the measure of its supplement. a. Write an equation that shows the relationship between m RQS and m SQT. b. Solve the equation and write a reason for each step. c. Find m RQT. Explain how you got your answer. 9
77. Given the following statements, can you conclude that Becky plays basketball on Wednesday night? If so, state which law justifies the conclusion. If not, write no conclusion. (1) If it is Wednesday night, Becky goes to the gym. (2) If Becky goes to the gym, she plays basketball. 78. In the figure shown, m AED = 115. True or False: AEB and AED are vertical angles and m AEB = 65. 84. Given: SQ bisects RST. Find QR if UT = 16 and UQ = 30. (not drawn to scale) 79. Name an angle supplementary to 2 in the figure below. 80. Tell whether the converse of the following statement is True or False. If it is false, give a counterexample. "If a number is even, then it is a multiple of 4." 81. Use the given angle measures to decide whether lines a and b are parallel. Write Yes or No. m 3 = 95, m 6 = 95 85. Given: SQ bisects RST. Find QR if UT = 35 and UQ = 120. (not drawn to scale) 86. Would HL, ASA, SAS, AAS, or SSS be used to justify that the pair of triangles is congruent? 82. The midpoint of QR is MÊ ËÁ 1, 7 ˆ. One endpoint is QÊ ËÁ 7,9 ˆ. Find the coordinates of the other endpoint. 83. The measure of an angle is 22 less than the measure of its supplement. Find the measure of the angle and the measure of its supplement. 87. Would HL, ASA, SAS, AAS, or SSS be used to justify that the pair of triangles is congruent? 88. ΔTRI ΔANG. Also T G. What type of triangle is ΔTRI? Explain. 10
89. HK and JL are angle bisectors. Which triangle congruence theorem or postulate could you use to prove that ΔHLJ ΔKLJ? Explain your answer. 93. Find the measure of the missing angle. 94. Find the value of x. 90. Which pair of lines is parallel if 1 is congruent to 7? Justify your answer. 95. Find the sum of the measures of the interior angles in the figure. 91. Which pair of lines is parallel if 4 is congruent to 2? Justify your answer. 96. What is the measure of each interior angle in a regular octagon? 92. a. Plot the following points in a coordinate plane: W (-3, -3), X (1, -6), Y (5, -3), Z (1, 0), b. Is WX congruent to YZ? Explain. c. Is there another pair of congruent segments? If so, name the segments and explain why they are congruent. 97. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 2340. 98. What is the measure of each exterior angle in a regular pentagon? 99. Given: Trapezoid ABCD with midsegment EF. If EF = 22 and DC = 26, find the length of AB. 11
Proofs 100. Given: ED EC; BD BC; ED BC Prove: ΔCED ΔDBC Statements Reasons 101. Writing: Write a paragraph proof to show that 4 and 5 are supplementary if 3 2. Statements Reasons 12
102. Complete the reasons of this proof. Given: AE DC; AB DB Prove: ΔABE ΔDBC Statements Reasons 1. AE Ä DC; AB DB 1. 2. A D 2. 3. ABE DBC 3. 4. ΔABE ΔDBC 4. 103. Provide the reasons for each statement in the proof. Given: m 1 = m 3 Prove: m AFC = m DFB Statement Reason m 1 = m 3? m 1 + m 2 = m 3 + m 2? m 1 + m 2 = m AFC, m 3 + m 2 = m DFB? m AFC = m DFB? 104. Given: BD is the median to AC, AB BC Prove: CBD ABD Statements Reasons 13
105. Given: ΔABC is an equilateral triangle; D is the midpoint of AC Statements Reasons Prove: ΔABD ΔCBD 106. Given: PR = 1 2 PT Statements Reasons Prove: R is the midpoint of PT 107. Given: ΔABF ΔDEC and FB Ä EC Prove: BCEF is a parallelogram. 108. Given: VU ST and SV TU Prove: VX = XT 14
109. Draw a figure in the coordinate plane and write a two-column coordinate proof. Given: Quadrilateral ABCD with A( 5, 0), B(3, 1), C(7, 3), D( 1, 2) Prove: ABCD is a parallelogram. 113. ΔABC ΔCDA. What special type of quadrilateral is ABCD? Write a paragraph proof to support your conclusion. 110. Use the Distance Formula to determine whether ABCD below is a parallelogram. 111. Given: ABCD is a rhombus. Prove: ΔACB ΔCAD 112. In the diagram, m BAC = 30, m DCA = 110, BCA DAC, and AC BD. Is enough information given to show that quadrilateral ABCD is an isosceles trapezoid? Explain. 15
ID: A Geometry First Semester Final Exam Review Answer Section MULTIPLE CHOICE 1. ANS: A STA: MI 9-12.G1.1.2 2. ANS: B STA: MI 9-12.L1.2.1 3. ANS: B STA: MI 9-12.G1.1.1 MI 9-12.G1.2.1 MI 9-12.G1.2.2 4. ANS: C 5. ANS: D STA: MI 9-12.G2.3.1 MI 9-12.G2.3.2 6. ANS: A STA: MI 9-12.G2.3.1 MI 9-12.G2.3.2 7. ANS: D STA: MI 9-12.A1.2.1 MI 9-12.A1.2.3 8. ANS: C STA: MI 9-12.G1.2.5 9. ANS: C STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 10. ANS: B STA: MI 9-12.G2.3.1 MI 9-12.G2.3.2 11. ANS: D STA: MI 9-12.G1.1.1 MI 9-12.G1.2.1 MI 9-12.G1.2.2 12. ANS: B STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 MI 9-12.G1.3.1 13. ANS: A STA: MI 9-12.G1.3.1 14. ANS: D STA: MI 9-12.G1.2.5 15. ANS: A STA: MI 9-12.G1.1.1 16. ANS: B STA: MI 9-12.G1.2.5 17. ANS: D STA: MI 9-12.G1.2.5 18. ANS: D STA: MI 9-12.G1.1.2 19. ANS: B 20. ANS: C 21. ANS: C STA: MI 9-12.A1.2.9 MI 9-12.A2.4.4 22. ANS: C STA: L4.3.2 23. ANS: D 24. ANS: C STA: MI 9-12.G1.1.1 25. ANS: C 26. ANS: A 27. ANS: A STA: MI 9-12.G1.1.2 28. ANS: B STA: MI 9-12.G1.3.1 29. ANS: B STA: MI 9-12.L4.3.3 30. ANS: A 31. ANS: A STA: MI 9-12.A1.2.1 MI 9-12.A1.2.3 MI 9-12.G1.1.1 32. ANS: A STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 33. ANS: D STA: MI 9-12.A1.2.9 MI 9-12.A2.4.4 34. ANS: C STA: MI 9-12.L1.1.6 MI 9-12.G1.2.3 MI 9-12.G1.2.5 MI 9-12.G1.3.1 35. ANS: B 36. ANS: A STA: MI 9-12.A2.4.4 37. ANS: D STA: MI 9-12.G1.1.2 38. ANS: C 39. ANS: C STA: MI 9-12.G1.5.2 40. ANS: B STA: MI 9-12.G1.4.1 MI 9-12.G1.4.2 MI 9-12.G1.4.3 1
ID: A 41. ANS: D STA: MI 9-12.G1.1.2 MI 9-12.G1.2.1 MI 9-12.G1.2.2 MI 9-12.G1.4.1 MI 9-12.G1.4.2 MI 9-12.G1.4.3 SHORT ANSWER 42. ANS: 7 4 STA: MI 9-12.A1.2.9 MI 9-12.A2.4.4 43. ANS: 144 STA: MI 9-12.G1.1.1 44. ANS: (4, 1) STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 MI 9-12.G1.4.2 45. ANS: If a number is even, then it is divisible by two. STA: MI 9-12.L4.2.4 46. ANS: m RPQ = 21 and m OPQ = 46 STA: MI 9-12.A1.2.1 47. ANS: 68 STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 MI 9-12.G1.3.1 MI 9-12.G2.3.1 MI 9-12.G2.3.2 48. ANS: 4 < x < 24 49. ANS: 39 < x < 95 50. ANS: 22 51. ANS: 32 STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 52. ANS: 113 STA: MI 9-12.G1.1.1 2
ID: A 53. ANS: invalid; converse error (The figure could have been a triangle.) STA: MI 9-12.L4.1.2 MI 9-12.L4.2.1 MI 9-12.L4.2.3 MI 9-12.L4.2.4 MI 9-12.L4.3.1 MI 9-12.L4.3.3 54. ANS: Ahmed will go to Belize, and Jake will go with him. STA: MI 9-12.L4.1.1 MI 9-12.L4.1.3 MI 9-12.L4.3.3 55. ANS: invalid; inverse error STA: MI 9-12.L4.1.2 MI 9-12.L4.2.3 MI 9-12.L4.2.4 MI 9-12.L4.3.1 MI 9-12.L4.3.3 56. ANS: If a number is even, then it is divisible by two. STA: MI 9-12.L4.2.4 57. ANS: neither STA: MI 9-12.A1.2.9 MI 9-12.A2.4.4 58. ANS: hypothesis: tomorrow is Friday, conclusion: today is Thursday STA: MI 9-12.L4.3.1 59. ANS: m QOP = 16 and m NOP = 44 STA: MI 9-12.A1.2.1 60. ANS: The life she saves may be her own. STA: MI 9-12.L4.1.1 MI 9-12.L4.1.3 MI 9-12.L4.3.3 61. ANS: 2 STA: MI 9-12.A1.2.1 MI 9-12.G1.1.1 62. ANS: 50. Since 2 and 3 form a linear pair, they are supplementary and the sum of their measures is 180. So, m 2 = 180 140 = 40. Since 1 and 2 are complementary, the sum of their measures is 90. So, m 1 = 50. STA: MI 9-12.L1.1.3 MI 9-12.L1.1.4 MI 9-12.L1.3.1 MI 9-12.A1.2.4 MI 9-12.A1.2.5 MI 9-12.A1.2.6 MI 9-12.A1.2.7 MI 9-12.A1.2.8 MI 9-12.G1.1.1 MI 9-12.G1.1.2 63. ANS: False; sample counterexample: 6 STA: MI 9-12.L4.1.3 MI 9-12.L4.2.1 MI 9-12.L4.2.4 MI 9-12.L4.3.2 3
ID: A 64. ANS: 66, 64, 50 (x = 33) STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 65. ANS: x = 82 STA: MI 9-12.G1.2.1 MI 9-12.G1.2.2 66. ANS: A right triangle STA: MI 9-12.G1.2.5 MI 9-12.G2.3.1 67. ANS: 12 STA: MI 9-12.G1.3.1 MI 9-12.G2.3.1 MI 9-12.G2.3.2 68. ANS: AB = 21, BC = 35 STA: MI 9-12.A1.2.1 MI 9-12.G1.1.5 69. ANS: ( 7, 9) STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 MI 9-12.G1.4.2 70. ANS: yes STA: MI 9-12.L4.2.1 71. ANS: hypothesis: today is Tuesday, conclusion: yesterday was Monday STA: MI 9-12.L4.3.1 72. ANS: 170 13.0 STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 MI 9-12.G1.4.2 MI 9-12.G2.3.4 73. ANS: Ê c + e 2, d + b ˆ ËÁ 2 74. ANS: The angle measures 74 and the supplement measures 106. 75. ANS: CB 4
ID: A 76. ANS: a. 4x 2 = x + 13 b. x = 5. 4x 2 = x + 13. 4x = x + 15, Addition Property of Equality; 3x = 15, Subtraction Property of Equality; x = 5, Division Property of Equality. c. m RQT = 36. If x = 5, then m RQS = 4(5) 2 = 18 and m SQT = 5 + 13 = 18. m RQT = m RQS + m SQT = 18 + 18 = 36. STA: MI 9-12.L1.1.1 MI 9-12.L1.1.3 MI 9-12.A1.2.3 MI 9-12.A1.2.4 MI 9-12.A1.2.5 MI 9-12.A1.2.6 MI 9-12.A1.2.8 77. ANS: yes STA: MI 9-12.L4.2.1 78. ANS: False 79. ANS: 1 or 3 STA: MI 9-12.G1.1.1 80. ANS: True STA: MI 9-12.L4.1.3 MI 9-12.L4.2.1 MI 9-12.L4.2.4 MI 9-12.L4.3.2 81. ANS: No 82. ANS: (-9, 5) STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 MI 9-12.G1.4.2 83. ANS: The angle measures 79 and the supplement measures 101. 84. ANS: 34 STA: MI 9-12.L1.1.6 MI 9-12.G1.2.3 MI 9-12.G1.2.5 MI 9-12.G1.3.1 85. ANS: 125 STA: MI 9-12.L1.1.6 MI 9-12.G1.2.3 MI 9-12.G1.2.5 MI 9-12.G1.3.1 86. ANS: AAS STA: MI 9-12.G2.3.1 MI 9-12.G2.3.2 5
ID: A 87. ANS: SAS STA: MI 9-12.G2.3.1 MI 9-12.G2.3.2 88. ANS: Isosceles. ΔTRI ΔANG., so TR AN. Then, since T G, RI AN by the Converse of the Isosceles Triangle Theorem. So, TR IR since congruence of segments is transitive. Therefore, ΔABC is isosceles. STA: MI 9-12.L1.1.3 MI 9-12.A1.2.4 MI 9-12.A1.2.5 MI 9-12.A1.2.6 MI 9-12.A1.2.7 MI 9-12.A1.2.8 MI 9-12.G1.2.2 89. ANS: ASA Congruence Postulate. Since HK and JL are angle bisectors, HLJ KLJ and HJL KJL. Since congruence of segments is reflexive, JL JL. Since you know that 2 pairs of angles and the included Side are congruent, you can use the ASA Congruence Postulate to prove the triangles are congruent. STA: MI 9-12.G1.1.1 MI 9-12.G1.4.2 MI 9-12.G2.3.1 MI 9-12.G2.3.2 90. ANS: c and d STA: MI 9-12.G1.1.2 91. ANS: c and d STA: MI 9-12.G1.1.2 92. ANS: a. See diagram below. b. Yes, because each has a length of 5. c. No. Sample answer: WY XZ because WY = 8 and XZ = 6. 93. ANS: 102 94. ANS: 53 STA: MI 9-12.G1.5.2 95. ANS: 720 6
ID: A 96. ANS: 135 STA: MI 9-12.G1.5.2 97. ANS: 15 98. ANS: 72 STA: MI 9-12.G1.5.2 99. ANS: 18 STA: MI 9-12.G1.4.1 MI 9-12.G1.4.2 MI 9-12.G1.4.3 OTHER 100. ANS: Statements Reasons 1. ED EC 1. Given 2. CED is a rt 2. If 2 segments are, they form rt s. 3. BD BC 3. Given 4. DBC is a rt 4. If 2 segments are, they form rt s. 5. ED BC 5. Given 6. DC CD 6. Reflexive 7. ΔCED ΔDBC 7. HL Congruence Theorem STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 101. ANS: Sample answer: Given: 3 2; Prove: 4 and 5 are supplementary From the given we know that 3 2. Since 2 and 5 are vertical angles, they are congruent. If 3 2 and 2 5, then 3 5 by the Transitive Property. By the definition of linear pair, 4 and 3 are a linear pair and therefore are supplementary. Since 3 5, 4 and 5 are supplementary. STA: MI 9-12.G1.4.2 7
ID: A 102. ANS: Statements Reasons 1. AE Ä DC; AB DB 1. Given 2. A D 2. If 2 parallel lines are intersected by a transversal, then alternate interior angles are congruent. 3. ABE DBC 3. Vertical angles are congruent. 4. ΔABE ΔDBC 4. ASA Postulate 103. ANS: Statement m 1 = m 3 m 1 + m 2 = m 3 + m 2 m 1 + m 2 = m AFC, m 3 + m 2 = m DFB m AFC = m DFB 104. ANS: 1. BD is the median to AC, AB BC 1. Given Reason Given Addition property of equality Angle addition postulate Substitution property of equality 2. AD DC 2. Definition of a median 3. BD BD 3. Reflexive 4. ΔADB ΔCDB 4. SSS 5. CBD ABD 5. Corresponding parts of Δs are STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 8
ID: A 105. ANS: Statements ΔABC is an equilateral triangle AB CB D is the midpoint of AC AD CD BD BD ΔABD ΔCBD Reasons Given Definition of equilateral Given Definition of midpoint Reflexive Property of Congruence SSSCongruence Postulate STA: MI 9-12.A1.2.3 MI 9-12.A1.2.4 MI 9-12.A1.2.5 MI 9-12.A1.2.6 MI 9-12.A1.2.8 MI 9-12.G1.1.3 MI 9-12.G1.1.5 MI 9-12.G2.3.1 MI 9-12.G2.3.2 106. ANS: Statement Reason 1. PR = 1 2 PT 1. Given 2. 2PR = PT 2. Multiplication property of equality 3. PR + PR = PT 3. Distributive property 4. PT = PR + RT 4. Segment Addition postulate 5. PR + PR = PR + RT 5. Transitive property of equality 6. PR =RT 6. Subtraction property of equality 7. R is the midpoint of PT 7. Definition of midpoint STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 107. ANS: 1. ΔABF ΔDEC 1. Given 2. BF EC 2. Corresponding Parts of Δ are. 3. FB Ä EC 3. Given 4. BCEF is a parallelogram. 4. If 1 pair of opposite sides are and, then the quadrilateral is a parallelogram. STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 9
ID: A 108. ANS: 1. VU ST and SV TU 1. Given 2. STUV is a parallelogram. 3. VX = XT 2. If both pairs of opp. sides of a quad. are, then the quad. is a parallelogram. 3. The diagonals of a parallelogram bisect each other. STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 109. ANS: 1. Quadrilateral ABCD with AÊ ËÁ 5, 0 ˆ, BÊ Ë Á 3, 1 ˆ, C Ê Ë Á7, 3 ˆ, D Ê ËÁ 1, 2 ˆ 2. slope of AB = 1 0 3 ( 5) = 1 8 slope of BC = 3 1 7 3 = 1 2 slope of CD = 2 3 1 7 = 1 8 slope of AD = 0 2 5 ( 1) = 1 2 1. Given 2. Definition of slope 3. AB Ä DC, AD Ä BC 3. Lines with = slopes are Ä. 4. ABCD is a parallelogram. 4. Definition of a parallelogram STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 MI 9-12.G2.3.4 10
ID: A 110. ANS: If ABCD is a parallelogram, then AB = DC. Since AB = 37 and DC = 40, ABCD is not a parallelogram. STA: MI 9-12.A1.2.9 MI 9-12.G1.1.5 MI 9-12.G1.4.1 MI 9-12.G1.4.2 MI 9-12.G1.4.3 MI 9-12.G2.3.4 111. ANS: 1. ABCD is a rhombus. 1. Given 2. ABCD is a parallelogram. 2. Definition of a rhombus 3. AB CD; BC AD 3. Opposite sides of a parallelogram are congruent. 4. AC AC 4. Reflexive Property 5. ΔACB ΔCAD 5. SSSCongruence Postulate STA: MI 9-12.G1.4.2 MI 9-12.G2.3.2 112. ANS: Yes, enough information is given to show ABCD is an isosceles trapezoid. ABCD is a trapezoid because BCA DAC so BC Ä AD. BAC and DCA are not congruent so BA is not parallel to CD. By definition ABCD is a trapezoid. The diagonals of trapezoid ABCD are congruent because AC BD. So, ABCD is an isosceles trapezoid by Theorem 8.16. STA: MI 9-12.G1.4.1 MI 9-12.G1.4.2 113. ANS: ABCD is a parallelogram. Since ΔABC ΔCDA and corresponding parts of congruent triangles are congruent, BAC DCA and BCA DAC. Therefore, AB Ä CD and AD Ä CB and ABCD is a parallelogram. OR Since ΔABC ΔCDA and corresponding parts of congruent triangles are congruent, AB CD and AD CB. Since both pairs of opposite sides are congruent, ABCD is a parallelogram. STA: MI 9-12.G1.4.1 MI 9-12.G1.4.2 MI 9-12.G1.4.3 MI 9-12.G2.3.2 11