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<strong>Answers</strong> to Exercises LESSON 5.6 1. Sometimes true; it is true only if the parallelogram is a rectangle. 2. Always true; by the definition of rectangle, all the angles are congruent. By the Quadrilateral Sum Conjecture and division, they all measure 90°, so any two angles in a rectangle, including consecutive angles, are supplementary. 3. Always true by the Rectangle Diagonals Conjecture. 4. Sometimes true; it is true only if the rectangle is a square. true true 5. Always true by the Square Diagonals Conjecture. 6. Sometimes true; it is true only if the rhombus is equiangular. true false false false 7. Always true; all squares fit the definition of rectangle. 8. Always true; all sides of a square are congruent and form right angles, so the sides become the legs of the isosceles right triangle and the diagonal is the hypotenuse. 9. Always true by the Parallelogram Opposite Angles Conjecture. 10. Sometimes true; it is true only if the parallelogram is a rectangle. Consecutive angles of a parallelogram are always supplementary, but are congruent only if they are right angles. 11. 20 12. 37° 13. 45°, 90° 14. DIAM is not a rhombus because it is not equilateral and opposite sides are not parallel. 15. BOXY is a rectangle because its adjacent sides are perpendicular. 16. Yes. TILE is a rhombus, and a rhombus is a parallelogram. 68 ANSWERS TO EXERCISES 17. 18. Constructions will vary. B A 19. one possible construction: I E V L O E K P S E 20. Converse: If the diagonals of a quadrilateral are congruent and bisect each other, then the quadrilateral is a rectangle. Given: Quadrilateral ABCD with diagonals AC BD. AC and BD bisect each other Show: ABCD is a rectangle A 8 E B D C 1 2 3 7 6 5 4 B A E Because the diagonals are congruent and bisect each other, AE BE DE EC.Using the Vertical Angles Conjecture, AEB CED and BEC DEA.So AEB CED and AED CEB by SAS. Using the Isosceles Triangle Conjecture and CPCTC, 1 2 5 6, and 3 4 7 8. Each angle of the quadrilateral is the sum of two angles, one from each set, so for example, mDAB m1 m8. By the addition property of equality, m1 m8 m2 m3 m5 m4 m6 m7. So mDAB mABC mBCD mCDA. So the quadrilateral is equiangular. Using 1 5 and the Converse of AIA, AB CD. Using 3 7 and the Converse of AIA, BC AD . Therefore ABCD is an equiangular parallelogram, so it is a rectangle. 21. If the diagonals are congruent and bisect each other, then the room is rectangular (converse of the Rectangle Diagonals Conjecture). K
22. The platform stays parallel to the floor because opposite sides of a rectangle are parallel (a rectangle is a parallelogram). 23. The crosswalks form a parallelogram: The streets are of different widths, so the crosswalks are of different lengths. The streets would have to meet at right angles for the crosswalks to form a rectangle. The corners would have to be right angles and the streets would also have to be of the same width for the crosswalk to form a square. 24. Place one side of the ruler along one side of the angle. Draw a line with the other side of the ruler. Repeat with the other side of the angle. Draw a line from the vertex of the angle to the point where the two lines meet. 25. Rotate your ruler so that each endpoint of the segment barely shows on each side of the ruler. Draw the parallel lines on each side of your ruler. Now rotate your ruler the other way and repeat the process to get a rhombus. The original segment is one diagonal of the rhombus. The other diagonal will be the perpendicular bisector of the original segment. 26. See flowchart below. 27. Yes, it is true for rectangles. Given: 1 2 3 4 Show: ABCD is a rectangle By the Quadrilateral Sum Conjecture, m1 m2 m3 m4 360°. It is given that all four angles are congruent, so each angle measures 26. (Lesson 5.6) 1 2 3 QU AD Given QD AU ? ? Given SSS DU DU Same segment 4 QUD ADU 8 Given 5 1 2 3 4 ? CPCTC QU UA AD DQ 90°. Because 4 and 5 form a linear pair, m4 m5 180°. Substitute 90° for m4 and solve to get m5 90°. By definition of congruent angles, 5 3, and 5 and 3 are alternate interior angles, so AD BC by the Converse of the Parallel Lines Conjecture. Similarly, 1 and 5 are congruent corresponding angles, so AB CD by the Converse of the Parallel Lines Conjecture. Thus, ABCD is a parallelogram by the definition of parallelogram. Because it is an equiangular parallelogram, ABCD is a rectangle. 28. a 54°, b 36°, c 72°, d 108°, e 36°, f 144°, g 18°, h 48°, j 48°, k 84° 29. possible answers: (1, 0); (0, 1); (1, 2); (2, 3) 30. y 8 6 x 9 8 9 or 8x 9y 86 7 31. y x 10 12 5 or 7x 10y 24 32. velocity 1.8 mi/h; angle of path 106.1° clockwise from the north 60 1.5 mi/h 2 mi/h 6 QU AD 7 QUAD is a QD AU parallelogram Converse of the Parallel Lines Conjecture 9 Definition of rhombus Definition of parallelogram QUAD is a rhombus ? ANSWERS TO EXERCISES 69 <strong>Answers</strong> to Exercises
Page 1 and 2:
Answers to Exercises 0 CHAPTER 0
Page 3 and 4:
LESSON 0.3 1. Design with reflectio
Page 5 and 6:
LESSON 0.5 1. possible answers: Sco
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CHAPTER 0 REVIEW 1. possible answer
Page 9 and 10:
USING YOUR ALGEBRA SKILLS 1 1. (3,
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34. B 35. MY; CK; mI 36. SEU; EUO;
Page 13 and 14:
20. false 21. true 22. false; Thoug
Page 15 and 16:
LESSON 1.5 1. D 2. A 3. C 4. B 5. C
Page 17 and 18: LESSON 1.7 1. Answers will vary. Sa
Page 19 and 20: LESSON 1.8 1. 2. 3. 4. 5. 6. 7. 8.
Page 21 and 22: CHAPTER 1 REVIEW 1. true 2. False;
Page 23 and 24: Answers to Exercises CHAPTER 2 •
Page 25 and 26: 8. 8n is steeper; the coefficient o
Page 27 and 28: LESSON 2.4 1. inductive; deductive
Page 29 and 30: LESSON 2.6 1. 63° 2. 90° 3. no 4.
Page 31 and 32: CHAPTER 2 REVIEW 1. poor inductive
Page 33 and 34: 1. 2. 3. 4. 5. A B Q D C D 1 CD 2 E
Page 35 and 36: 1. LESSON 3.3 The answer depends on
Page 37 and 38: 1. D 2. F 3. A 4. C 5. E 6. 7. 8. 9
Page 39 and 40: LESSON 3.5 1. 2. 3. Construct a seg
Page 41 and 42: LESSON 3.6 1. Sample description: C
Page 43 and 44: 1. incenter LESSON 3.7 Because the
Page 45 and 46: LESSON 3.8 1. The center of gravity
Page 47 and 48: 42. true 43. False; any non-acute t
Page 49 and 50: LESSON 4.2 1. 79° 2. 54° 3. 107.5
Page 51 and 52: 1. yes 2. no 3. no 4 5 9 5 6 12 LES
Page 53 and 54: LESSON 4.5 1. If two angles and the
Page 55 and 56: 1. See flowchart below. 2. See flow
Page 57 and 58: 1. 6 2. 90°; 18° 3. 45° 4. See f
Page 59 and 60: CHAPTER 4 REVIEW 1. Their rigidity
Page 61 and 62: Answers to Exercises 5 CHAPTER 5
Page 63 and 64: LESSON 5.3 1. 64 cm 2. 21°; 146°
Page 65 and 66: 1. 34 cm; 27 cm 2. 132°; 48° 3. 1
Page 67: 1. 2. 3. (0, 4) y y y (0, 6) USING
Page 71 and 72: 7. 2 RE AB 3 BR EA 4 AR EB
Page 73 and 74: 20. Speed: 901.4 km/h. Direction:
Page 75 and 76: LESSON 6.2 1. 165°, definition of
Page 77 and 78: 1 OR CD ? 5 OC OD ? All radii o
Page 79 and 80: LESSON 6.5 1. d 5 cm 2. C 10 cm 3
Page 81 and 82: 1. 1 2 ,3 USING YOUR ALGEBRA SKILL
Page 83 and 84: CHAPTER 6 REVIEW 1. Answers will va
Page 87 and 88: 15. possible answer: HIKED 16. one,
Page 89 and 90: LESSON 7.4 1. Answers will vary. 2.
Page 91 and 92: LESSON 7.6 1. Answers will vary. 2.
Page 93 and 94: 1. parallelograms 2. parallelograms
Page 95 and 96: CHAPTER 7 REVIEW 1. true 2. true 3.
Page 97 and 98: 1. 20 cm 2 2. 49.5 m 2 3. 300 squar
Page 99 and 100: USING YOUR ALGEBRA SKILLS 8 1. x 2
Page 101 and 102: LESSON 8.5 1. 9 in 2 2. 49 cm 2 3.
Page 103 and 104: LESSON 8.7 1. 150 cm 2 2. 4070 cm 2
Page 107 and 108: USING YOUR ALGEBRA SKILLS 9 1. 6 2.
Page 109 and 110: LESSON 9.4 1. No. The space diagona
Page 111 and 112: LESSON 9.6 1. 18 cm2 56.5 cm2 2. (
Page 113 and 114: 32. 4 in./s 12.6 in./s 33. true 34
Page 115 and 116: 36. See table below. Possible answe
Page 117 and 118: LESSON 10.3 1. 192 cm3 2. 84 cm3 2
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LESSON 10.5 1. 675 cm3 2. 36 cm3 1
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LESSON 10.7 1. V 972 cm3 3054 cm3
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CHAPTER 10 REVIEW 1. They have the
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1. A 2. B 3. possible answer: 3. po
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LESSON 11.3 1. 16 m 2. 4 ft 3 in. 3
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1. 18 cm 2. 12 cm 3. 21 cm 4. 15 cm
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1. 1715 cm 3 LESSON 11.6 2. 16 cm,
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Use segment addition to show that e
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Answers to Exercises CHAPTER 12 •
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LESSON 12.3 1. 329 cm 2 2. 4 cm 2 3
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LESSON 12.5 1. approximately 12.7 m
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13. f(x) 1 2 (x3)2 2; A vertical s
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Answers to Exercises CHAPTER 13 •
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LESSON 13.2 1. Linear Pair Postulat
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LESSON 13.3 1. Case 1: P is colline
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1. A LESSON 13.4 Use x to represent
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LESSON 13.5 1. D: Paris is in Franc
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8. Use the Inscribed Angle Theorem
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11. CA Postulate 12. 13. Perpendicu
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8. Task 1: Given: A rectangle with
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CHAPTER 13 REVIEW 1. False. The qua
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