Name: ________________________ Class: ___________________ Date: __________ ID: A
Geometry First Semester Final Exam Review
Multiple ChoiceIdentify the choice that best completes the statement or answers the question.
1. Find m∠1 in the figure below. PQ←⎯⎯→ and RS←⎯⎯→ are parallel.
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. < b. > c. =
d. not enough information 3. Find the value of x:
a. 88°
4. If m∠AOC = 48° and m∠BOC = 21°, then what is the measure of ∠AOB?
a. 24° b. 32° c. 27° d. 29°
5. ΔABD ≅ ΔCBD. Name the theorem or postulate that justifies the congruence.
a. HL b. SAS c. AAS d. ASA
6. ΔABD ≅ ΔCBD. Name the theorem or postulate that justifies the congruence.
a. AAS b. SAS c. HL
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 4y − 12ÊËÁÁ ˆ¯˜˜° and the measure of ∠2 is 48°. (The figure may not be drawn to scale.)
a. 22 b. 69 c. 23 d. 36
8. In the diagram below, KF←⎯⎯→ is the perpendicular bisector of GH. Then ∠KGF ≅ .
a. ∠FKG b. ∠KF c. ∠KHF d. ∠KFH
9. The distance between points A and B is _______.
a. 13
b. 11
10. Which postulate or theorem can be used to determine the length of RT?
a. ASA Congruence Postulate b. AAS Congruence Theorem c. SSS Congruence Postulate d. SAS Congruence Postulate 11. Find the value of x:
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°
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?
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.
a. ∠COB b. ∠DOB
c. ∠DOE or ∠AOC d. ∠DOC or ∠AOE
Refer to the figure below for the next two questions.
Given: AF ≅ FC, ∠ABE ≅ ∠EBC 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. 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
20. If AB = 14 and AC = 27, find the length of BC.
a. 14 b. 41 c. 13 d. 3
21. Which best describes the relationship between
Line 1 and Line 2?
Line 1 passes through 2Ê , −7
ËÁÁ ˆ¯˜˜ and 4, −3ÊËÁÁ ˆ¯˜˜
Line 2 passes through ÊËÁÁ−4,9ˆ¯˜˜ and −2, 13ÊËÁÁ 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?
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. 18°
b. 72° c. 108° d. 28°
25. Refer to the figure.
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. alternate exterior angles b. consecutive interior angles c. corresponding angles d. alternate interior angles
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?
a. 138° b. 69° c. 42° d. 48°
29. In the figure shown, m∠CED = 63°. Which of the following statements is false?
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 31. Solve for x. 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
33. A line L1 has slope 4
9. The line that passes through which of the following pairs of points is parallel to L1?
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
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?
a. ∠8 ≅ ∠2 b. ∠2 ≅ ∠6 c. ∠5 ≅ ∠3 d. ∠7 ≅ ∠4
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. 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°
40. For parallelogram PQLM below, if m∠PML = 83°, then m∠PQL =______ . a. m∠PQM b. 83° c. 97° d. m∠QLM
41. Find the value of the variables in the parallelogram. 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.
47. Line l is the perpendicular bisector of MN. Find m∠M.
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.
52. ∠1 and ∠2 are supplementary angles. ∠1 and ∠3 are vertical angles. m∠2=67°. Find m∠3.
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.
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. 61. Solve for x:
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.
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.
65. Solve for x.
66. If QR is an altitude of ΔPQR, what type of triangle is ΔPQR?
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.
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.
75. Identify the longest side of ΔABC.
76. QS⎯⎯→ bisects ∠RQT
m∠RQS = (4x − 2)° and m∠SQT = (x + 13)°.
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.
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°.
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°
82. The midpoint of QR is MÊËÁÁ−1, 7ˆ¯˜˜. One endpoint is Q 7, 9ÊËÁÁ 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.
84. Given: SQ⎯⎯→ bisects ∠RST . Find QR if UT = 16 and UQ= 30. (not drawn to scale)
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?
87. Would HL, ASA, SAS, AAS, or SSS be used to justify that the pair of triangles is congruent?
89. HK and JL are angle bisectors. Which triangle congruence theorem or postulate could you use to prove that ΔHLJ≅ ΔKLJ? Explain your answer.
90. Which pair of lines is parallel if ∠1 is congruent to ∠7? Justify your answer.
91. Which pair of lines is parallel if ∠4 is congruent to ∠2? Justify your answer.
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.
93. Find the measure of the missing angle.
94. Find the value of x.
95. Find the sum of the measures of the interior angles in the figure.
96. What is the measure of each interior angle in a regular octagon?
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
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 ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________
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 ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________
105.
Given: ΔABC is an
equilateral triangle; D is the midpoint of AC Prove: ΔABD ≅ ΔCBD Statements Reasons ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ 106. Given: PR= 1 2 PT
Prove: R is the midpoint of
PT Statements Reasons ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________ ______________________ _________________________
107. Given: ΔABF ≅ ΔDEC and FB Ä EC Prove: BCEF is a parallelogram.
108. Given: VU ≅ ST and SV ≅ TU Prove: VX = XT
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.
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.
113. ΔABC ≅ ΔCDA. What special type of
quadrilateral is ABCD? Write a paragraph proof to support your conclusion.
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.G1.2.59. 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
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°
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
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: c+ e 2 , d + b2 Ê Ë ÁÁÁÁ ÁÁ ˆ ¯ ˜˜˜˜ ˜˜ 74. ANS:
The angle measures 74° and the supplement measures 106°. 75. ANS:
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
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:
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 T heorem 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.
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 Reason
m∠1 = m∠3 Given
m∠1 + m∠2 = m∠3 + m∠2 Addition property of equality
m∠1 + m∠2 = m∠AFC, m∠3 + m∠2 = m∠DFB Angle addition postulate
m∠AFC = m∠DFB Substitution property of equality
104. ANS:
1. BD is the median to AC, AB ≅ BC 1. Given
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
ID: A
105. ANS:
Statements Reasons
ΔABC is an equilateral triangle Given
AB≅ CB Definition of equilateral
D is the midpoint of AC Given
AD≅ CD Definition of midpoint
BD≅ BD Reflexive Property of Congruence ΔABD ≅ ΔCBD 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. T ransitive 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
ID: A
108. ANS:
1. VU≅ ST and SV ≅ TU 1. Given
2. STUV is a parallelogram. 2. If both pairs of opp. sides of a quad. are ≅ , then the quad. is a parallelogram. 3. VX= XT 3. T he 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−5, 0 ËÁÁ ˆ¯˜˜, B 3ÊËÁÁ , 1ˆ¯˜˜, C 7, 3ÊËÁÁ ˆ¯˜˜, D −1, 2ÊËÁÁ ˆ¯˜˜ 1. Given 2. slope of AB= 1 − 0 3− −5( ) = 18 slope of BC= 3 − 1 7 − 3 = 12 slope of CD= 2 − 3 −1 − 7 = 18 slope of AD= 0 − 2 −5 − −1( ) = 12 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
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.
