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All rights reserved. Chapter 3 Answers Practice 3-1 Practice 3-4 1. corresponding angles 2. alternate interior angles 3. same-side interior angles 4. alternate interior angles 5. same-side interior angles 6. corresponding angles 7. 1 and 5, 2 and 6, 3 and 8, 4 and 7 8. 4 and 6, 3 and 5 9. 4 and 5, 3 and 6 10. m1 = 100, alternate interior angles; m2 = 100, corresponding angles or vertical angles 11. m1 = 75, 1. 125 2. 69 3. 143 4. 129 5. 140 6. 136 7. x = 35; y = 145; z = 25 8. a = 55; b = 97; c = 83 9. v = 118; w = 37; t = 62 10. 50 11. 88 12. m3 = 22; m4 = 22; m5 = 88 13. 57.1 14. 136 15. m1 = 33; m2 = 52 16. isosceles 17. obtuse scalene 18. right scalene 19. obtuse isosceles 20. equiangular equilateral alternate interior angles; m2 = 75, vertical angles or corresponding angles 12. m1 = 135, corresponding angles; m2 = 135, vertical angles 13. x = 103; 77°, 103° 14. x = 24; 12°, 168° 15. x = 30; 85°, 85° 16a. Alternate Interior Angles Theorem 16b. Vertical angles are congruent. 16c. Transitive Property of Congruence Practice 3-2 * ) * ) 1a. same-side interior 1b. QR 1c. TS 1d. * same-side interior 1e. Same-Side Interior Angles ) 1f. TS 1g. 3-5 2. l and m, Converse of Same-Side Interior Angles Theorem 3. none 4. BC and AD , Converse of Same-Side Interior Angles Theorem 5. RT and HU , Converse of Corresponding Angles Postulate 6. BH and CI, Converse of Corresponding Angles Postulate 7. a and b, Converse of Same-Side Interior Angles Theorem 8. 43 9. 90 10. 38 11. 100 12. 70 13. 48 Practice 3-5 1. x = 120; y = 60 2. n = 5137 3. a = 108; b = 72 4. 109 5. 133 6. 129 7. 129 8. 47 9. 127 10. 30 11. 150 12. 6 13. 5 14. 8 15. BEDC 16. FAE 17. FAE and BAE 18. ABCDE 19. 20 Practice 3-6 1. y = 13 x - 7 4. y 9. © Pearson Education, Inc., publishing as Pearson Prentice Hall. = 16 x - 23 y 6 4 2 y 4x 1 2 4 6 x y 1 x 2 3 2 4 6 x –6–4–2 –2 –4 –6 y 1 x 5 4 x 4 2 –6–4 –2 –2 –4 –6 2 4 6 x –6–4–2 –4 –6 y 15. 6 4 2 2 4 6 6 y 2x 7 4 2 –6–4 –2 –2 –4 –6 y yx1 12. y 6 4 2 –6–4–2 –2 –4 –6 13. 6. y = 45 x - 2 10. –6 11. 3. y = 7x - 18 8. y = -x + 6 y 6 4 2 –6–4–2 –2 l2 5. y 3 7. y = 4x - 13 Practice 3-3 1. True. Every avenue will be parallel to Founders Avenue, and therefore every avenue will be perpendicular to Center City Boulevard, and therefore every avenue will be perpendicular to any street that is parallel to Center City Boulevard. 2. Not necessarily true. No information has been given about the spacing of the streets. 3. True. The fact that one intersection is perpendicular, plus the fact that every street belongs to one of two groups of parallel streets, is enough to guarantee that all intersections are perpendicular. 4. True. Opposite sides of each block must be of the same type (avenue or boulevard), and adjacent sides must be of opposite type. 5. Not necessarily true. If there are more than three avenues and more than three boulevards, there will be some blocks bordered by neither Center City Boulevard nor Founders Avenue. 6. a ⊥ e 7. a || e 8. a || e 9. a ⊥ e 10. a ⊥ e 11. a || e 12. If the number of ⊥ statements is even, then 1 || n. If it is odd, then 1 ⊥ n. 13. The proof can instead use alternate interior angles or alternate exterior angles (if the angles are congruent, the lines are parallel) or same-side interior or same-side exterior angles (if the angles are supplementary, the lines are parallel). 14. It is possible. 2. y = -2x + 12 1 = -2 x - 14. 2 4 6 x y 6 y 2x 3 4 2 –6–4–2 –2 y 16. y 3x 5 2 4 6 x y 2 6 y x2 3 4 4 x 2 4 6 –6 –2 –2 –4 –6 2 4 6 x l1 l3 Geometry Chapter 3 Answers 39 Chapter 3 Answers 2 4 6 x –6–4–2 20. y 6 4 2 x 2 –6–4 2 4 6 x –2 –4 –6 21. 22. y –4 –6 2 4 6 x y 23. y 6 4 2 y – 3x 2 2 –6–4–2 –2 –4 –6 yx 2 4 6 x –4 –6 24. 2 4 6 x x 2.5 2 4 6 x –6–4–2 –2 –4 –6 25. y = –3x + 13 26. y = x + 4 27. y = 21 x - 3 1 1 28. y = -2 x - 2 29. y = 2x + 4 30. y = 13 x + 4 31. y = -51 x - 65 32. y = -6x + 45 33. x = 2; y = -11 34. x = 0; y = 2 35. x = -4; y = -4 36. x = -1; y = 8 37. y 6 4 (4, 0) 2 38. y (–2, 0) 6 4 2 6 x –6–4–2 2 –2 –4 –6 –8 –10 –12 (0, 12) –6 –4–2 –2 (0, 6) (9, 0) x 2 4 6 8 –4 –6 –2 –2 2 4 6 x –4 –6 45a. m = $0.10 45b. the amount of money the worker is paid for each box loaded onto the truck 45c. b = $3.90 45d. the base amount the worker is paid per hour 46. y = -25 x + 8 Practice 3-7 y 6 4 2 2 4 6 8 x y 6 4 2 6 4 (–3, 0) 2 –6 –6–4–2 y 5 44. 12 (0, 12) 10 6 4 2 2 4 6 x –6–4–2 –2 –6–4–2 –2 –4 –6 y 6 4 –6–4–2 –2 –4 –6 6 4 2 y 6 4 2 (0, 1) (8, 0) 2 4 6 x –6–4–2 –2 –4 (0, –4) –6 43. y y 2x 42. y 6 4 2 (4, 0) 6x –6–4–2 2 –2 1 –4 y 2 x 3 –6 –4 –6 19. 41. y 6 4 2 –6 –4 –2 2 4 6 x –2 –4 –6 (0, –1) 1. neither; 3 2 31, 3 ? 31 2 -1 2. perpendicular; 1 2 ? 2 = 1 3. parallel; = -23 4. parallel; 2 3 -1 = -1 5. perpendicular; y = 2 is a horizontal line, x = 0 is a vertical line 6. parallel; -12 = -12 7. neither; 1 2 81, 1 ? 18 2 -1 8. parallel; -23 = -23 9. perpendicular; -1 ? 1 = -1 10. neither; 21 2 5 1 5 7 7 2 -3, 2 ? –3 2 -1 11. neither; -3 2 -12 , -23 ? -12 2 -1 1 1 12. neither; 6 2 -5, 6 ? -5 2 -1 13. neither; 29 2 4, 92 ? 4 2 -1 14. parallel; 21 = 12 15. y = 32 x 4 16. y = -3 x + 24 y 17. y = -x - 3 6 4 18. y = 35 x + 6 2 19. y = 0 –6–4 L 4 6 x 20. y = 2x - 4 –4 21. y = 2x Practice 3-8 1.–3. 4.–6. Q 39. y 6 (0, 6) 4 2 –6–4–2 –2 –4 –6 40 2 4 6 x (6, 0) Answers 40. y 6 4 1 (– 2, 0) 2 (0, 2) –6–4–2 All rights reserved. 18. y 6 4 y 5x 4 2 T 2 4 6 x –4 –6 Geometry Chapter 3 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 17. (continued) Chapter 3 Answers (continued) 14. Sample: 7.–9. K b c 10. Sample: 15. All rights reserved. b a a 11. Sample: a b Reteaching 3-1 1a.–1b. Sample: c 60 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 12. 1c.–1d. Sample: b b 120 60 60 120 b 120 60 60 120 b 2. 110 7. 70 3. 70 8. 110 4. 110 5. 110 6. 70 Reteaching 3-2 13. 1. l6m 1 3 Given If 6 lines, then corresponding s are . a m3 = m2 = 180 Substitution c 2 and 3 are supplementary. Definition of supplementary m1 + m2 = 180 Angle Addition Postulate Geometry Chapter 3 Answers 41 (continued) 2. 2 3 Given 3 1 a6b Substitution If corresponding s, then lines are parallel. 2 1 Vertical s are . 15. y = 14 x + 21 17. y = -x + 1 Reteaching 3-7 1a. -2 1b. 12 2a. 14 3b. 0 4a. y = -2x + 4 4c. 2b. -4 3a. undefined 4b. y = 12 x + 32 y Reteaching 3-3 1. Draw a temporary line through A that is perpendicular to , then draw a second, permanent line through A that is perpendicular to the first. This line will be parallel to . Repeat for B. The two permanent lines will be parallel not only to but to each other (Theorem 3-9). 2. Draw two perpendicular lines through C, then for each one draw a line perpendicular to it that runs through D. The four lines will define a rectangle. 3. Draw a line that runs through both E and F, and then a second line at an acute or obtuse angle that runs through E. Draw a third line, parallel to the second and running through F, using the same procedure as in the Example and in Exercise 1. In the same way, draw a fourth line parallel to the first (at any desired distance from it). The four lines will define a parallelogram. 16. y = -34 x + 4 18. y = 1 4 2 2 O 2 4 x All rights reserved. Chapter 3 Answers 2 4 5b. y = 32 x - 4 5a. y = -32 x - 4 5c. y 4 Reteaching 3-4 2 1. ABD: mABD = 120, mADB = 30; CBE: 4 m3 = 60; m4 = 120 2 4 x 2 4 6a. y = 3x + 7 6c. 6b. y = -13 x - 3 y 4 2 Reteaching 3-5 1. 1 and 2 are interior angles; 3 and 4 are exterior angles. 2. m1 = 135; m2 = 90; m3 = 45; m4 = 90 3. 1 is an interior angle; 2 and 4 are exterior angles; 3 is neither. 4. m1 = 60; m2 = 120; 2 O 4 2 O 2 4 x 2 4 Reteaching 3-6 Check students’ graphs. 1. y = 2x - 6 2. y = 13 x 3. y = -x - 3 5 4. y = 6 x + 2 5. y = -12 x + 1 6. y = 1 7. y = -72 x + 10 8. y = -x + 1 9. y = 52 x + 1 10. y = 1 11. y = -2x - 6 12. x = -3 13. y = -3x + 10 14. y = 3x - 10 42 Answers 7. mJK = -1; mLM = -1; parallel 8. mJK = 32 ; mLM = 3 1 -2 ; perpendicular 9. mJK = -6; mLM = -15 ; neither 3 10. mJK = -2 ; mLM = 54; neither 11. mJK = 2; mLM = 1 1 -2; perpendicular 12. mJK = 5; mLM = 5; neither 13. mJK = 14; mLM = -14; parallel 14. mJK undefined; mLM = 0; perpendicular Geometry Chapter 3 © Pearson Education, Inc., publishing as Pearson Prentice Hall. mCBE = 120, mCEB = 30, mBCE = 30; BDE: mBDE = 60, mDBE = 60, mBED = 60 2. DBE and ABC are acute, equiangular, and equilateral; ABD and CBE are isosceles and obtuse; ACE, ADE, CED, and CAD are right and scalene. 3. PQT: mPTQ = 45, mPQT = 90; PQR: mPQR = 90, mQPR = 45, mQRP = 45; RQS: mRQS = 90, mQSR = 45; SQT: mSQT = 90, mQST = 45, mSTQ = 45 4. PT = TS = RS = PR = 40 mm; PQ = QT = QR = QS = 28 mm 5. PQT, PQR, RQS, SQT, PRS, PTS, PRT, and RST are right and isosceles. Chapter 3 Answers (continued) 8. Figures A and B ones, with precise control of the slant. would be easy, because the lines have no more than two or three different slants. Figure C would be harder, because lots of different slants are used to achieve the perspective effect. Reteaching 3-8 1. Sample: Z Enrichment 3-4 b X a 1. 48 2. 2880 3. 4320 4. Angles have measures of 20, 70, or 90; AC 6 MD 6 LE 6 KF 6 JG; BM6 CH 6 NK; AH 6 PG; CM 6 DL 6 EK 6 JF 6 IG. Y Enrichment 3-5 All rights reserved. 1. 2 2. 5 7. Number 2.–4. Check students’ work. Enrichment ) * 3-1 ) 1. OE * is #) to AB . 2. 3, 5, 1, 4, 2, 6 or 4, 5, 1, 3, 2, 6; OE ) is # to AB ; if two angles are congruent and supplementary, then each measures 90°. 3. 1 > 2; Law of Reflection 4. 2 > 3; Alternate Interior Angles Theorem 5. 3 > 4; Law of Reflection 6. 1 > 4; Transitive Property of Congruence 3. 9 1.–8. © Pearson Education, Inc., publishing as Pearson Prentice Hall. y 3 1 5 A 4 T 2 D 3. 33 4. 41 Enrichment 3-3 1. If the paper shifts during drawing, lines meant to be parallel will not be parallel, and lines meant to be perpendicular will not be perpendicular. 2. The rectangular shape gives the T-square two perpendicular guide edges on which to line up. 3. Theorem 3-10, which says that if two lines are perpendicular to the same line, they are parallel to each other. A line drawn with the T-square is perpendicular to the edge on which the Tsquare is lined up. Two lines drawn with the T-square lined up on the same edge will be parallel to each other. 4. a ⊥ d. If the T-square is used to draw a line perpendicular to a vertical edge of the rectangular backing surface, and then is used to draw a line perpendicular to a horizontal edge of the backing surface, then the two lines will be perpendicular. 5. Figure A would be easy; all the lines are parallel or perpendicular. Figures B and C would be hard to draw, since the lines are not all parallel or perpendicular to each other. 6. They will all be parallel to . 7. A drafting machine can be used to draw slanted lines, not just vertical and horizontal 6 L 5. Sample: H Because mBAC = 41, mCAF = 180 - 41 = 139; l 6 m because a pair of alternate interior angles are congruent. 6. CD and EF ; AK 7. Sample: CAB and IGH are corresponding angles related to parallel segments AB and GH . AK is the related transversal. 8. Sample: A, ADE, and AED form a triangle, so 180 - (43 + 76) = 61, so mAED = 61. Because AED and C are congruent corresponding angles, DE 6 BC by the Converse of the Corresponding Angles Postulate. Geometry Chapter 3 6. 27 Enrichment 3-6 Enrichment 3-2 2. 106 5. 20 3 4 5 6 7 8 9 ... n of sides Total 180 360 540 720 900 1080 1260 (n 2)180 degree measure Number of n(n 3) 0 2 5 9 14 20 27 2 diagonals 8 1. x = 11 4. 14 N B M x E K Y P C S 7 R RENE DESCARTES Enrichment 3-7 1. (-3, 4) 2. (-2, 3) 3. (3, 3) 4. (-2, -2) 5. (-1, -1) 6. (4, -1) 7. (1, 0) 8. (2, -3) 9. (-1, 2) 10. (1, -4) 11. (-3, -3) 12. (2, 3) 13. (3, -2) 14. (5, 0) 15. (0, 1) 16. (4, 3) y-axis 4 A 3 N 2 G 1 L 0 1 E 2 N I 3 4 L 4 3 2 1 0 R A Y E T N I O P 1 2 3 4 Answers 5 x-axis 43 Chapter 3 Answers (continued) pentagons Enrichment 3-8 1., 3., and 4. hexagons A C 2. scalene, acute triangle All rights reserved. B Activity 3: Analyzing 5. No; refer to the answer to Exercises 1, 3, and 4. 6. E Activity 4: Modeling F 7. isosceles, obtuse triangle Chapter Project Activity 1: Paper Folding 5; all triangles are right isosceles; yes. Activity 2: Exploring triangle ✔ Checkpoint Quiz 1 1. Converse of Corresponding Angles Postulate 2. Alternate Interior Angle Theorem 3. Same-Side Interior Angles Theorem 4. Corresponding Angles Postulate 5. Converse of Alternate Interior Angle Theorem 6. Vertical Angles Theorem 7. Converse of Corresponding Angles Postulate 8. Corresponding Angles Postulate 9. Converse of Same-Side Interior Angles Theorem 10. x = 50, y = 30, z = 65 ✔ Checkpoint Quiz 2 1. pentagon; 80 2. hexagon; x = 110; y = 98 3. quadrilateral; x = 94; y = 105 y y 4. 5. quadrilaterals 8 6 4 2 (2,0) 4 6 8 x 8642 2 2 4 6 8 (0,8) 44 Answers 8 6 4 (0,3) (1.2,0) 2 2 4 6 8 x 8642 2 4 6 8 Geometry Chapter 3 © Pearson Education, Inc., publishing as Pearson Prentice Hall. D Chapter 3 Answers (continued) y 6. Chapter Test, Form B 8 6 4 2 42 2 4 6 8 1. true 2. true 3. false 4. true 5. false 6. m1 = 62, m2 = 62, alternate interior angles 7. m1 = 85, m2 = 95, same-side interior angles 8. m1 = 75, m2 = 75, alternate interior angles y y 9. 10. (12,0) 2 4 6 8 10 12 x (0,8) 7. Parallel; slopes are the same. 8. neither 9. Perpendicular; the product of the slopes is -1. 10. neither Chapter Test, Form A All rights reserved. 1. true 6. false 2. true 7. true 3. false 8. true 4. false 5. true 9. Answers may vary. Sample: m1 = 125, Same-Side Interior Angles Theorem; m2 = 55, Alternate Interior Angles Theorem 10. Answers may vary. Sample: m1 = 60, Corresponding Angles Postulate then Angle Addition Postulate; m2 = 60, Same-Side Interior 11. Answers may vary. Sample: m1 = 85, Angles Theorem Alternate Interior Angles Theorem; m2 = 95, Same-Side 12. Answers may vary. Sample: Interior Angles Theorem m1 = 75, Corresponding Angles Postulate; m2 = 105, 13. Answers may vary. Sample: Angle Addition Postulate m1 = 91, Corresponding Angles Postulate and Same-Side Interior Angles Theorem; m2 = 89, Corresponding Angles 14. Answers may vary. Sample: m1 = 60, Postulate Alternate Interior Angles Theorem; m2 = 115, Same-Side Interior Angles Theorem 15. © Pearson Education, Inc., publishing as Pearson Prentice Hall. X 4 4 4 2 2 2 2 4 x 2 2 4 x 4 4 11. AD and WZ 12. AD and WZ 13. AD and WZ 14. AW and DZ 15. x = 92, y = 88 16. w = 31, x = 65, y = 65, z = 115 17. 1440 18. 45 19. neither 20. perpendicular 21. parallel 22. y = 3x - 1 23. y = 14 x - 3 24. y = -2x - 2 Alternative Assessment, Form C TASK 1: Scoring Guide a. t 1 2 4 3 5 8 6 7 a b b. Sample: 8 2; 8 4; 8 and 3 are supplementary. c. m2 = 75; m3 = 105, m4 = 75, m5 = 105, m6 = 75, m7 = 105, m8 = 75 m d. 7 and 4 are supp. Given 16. x 4 8 a6b Supp. of the same are . If alt. int s are , then lines are 6. 7 and 8 are supp. y 17. WA and XB 18. none 19. WZ and AB 20. none 21. WZ and AB 22. WZ and AB; AX and BY 23. x = 22; y = 120 24. x = 70; y = 60; z = 120 25. x = 35; y = 35; z = 55 26. 5940 27. 18 28. perpendicular 29. neither 30. parallel 31. y = 6x + 23 32. y = -21 x + 3 33. y = 13 x + 2 Geometry Chapter 3 Angle Add. Post. 3 Student draws an accurate diagram and supplies correct answers and a complete and accurate flow proof. 2 Student draws a figure or gives answers that contain minor errors. 1 Student draws a figure or gives answers that contain significant errors or omissions. 0 Student makes little or no attempt. Answers 45 (continued) TASK 2: Scoring Guide TASK 4: Scoring Guide Sample: Sample: Q A P Isosceles Triangles B Equilateral Triangles 3 Student constructs an accurate figure. 2 Student constructs a figure that contains minor errors or omissions. 1 Student constructs a figure that contains significant errors or omissions. 0 Student makes little or no attempt. TASK 3: Scoring Guide Sample: Triangles R 3 Student draws an accurate diagram. 2 Student draws a diagram that contains minor errors or omissions. 1 Student draws a diagram that contains significant errors or omissions. 0 Student makes little or no attempt. All rights reserved. Chapter 3 Answers Cumulative Review B A D C 1. D 2. G 3. C 4. F 5. D 6. J 7. A 8. F 9. C 10. H 11. D 12. J 13. B 14. G 15. C 16. G 17. C 18. Given 19. Same-Side Interior Angles Theorem 20. Corresponding Angles Postulate 21. Corresponding Angles Postulate 22. substitution 23. Check students’ work. 24. Sketches may vary. The right angle must be between the equal sides. 25. No; a triangle cannot have © Pearson Education, Inc., publishing as Pearson Prentice Hall. two sides equal and no sides equal at the same time. a. For the figure given, y = x and y = x + 3. The lines are parallel because both lines have a slope of 1. c. For ABC, the sum of the three angles is 180°. Minor discrepancies are the result of measurement error and rounding error. d. For the figure given, ABDC is not regular. By the distance formula, the sides are not congruent. 3 Student draws the figure accurately, writes correct equations, and reasons logically. 2 Student draws a figure, gives arguments, and writes equations that are mainly correct but may contain minor errors. 1 Student presents work with significant errors. 0 Student makes little or no attempt. 46 Answers Geometry Chapter 3