CHAPTER 4 Chapter Opener Chapter Readiness Quiz (p. 17) 1. D. H; PQ **** is horizontal, so subtract the x-coordinates. PQ 7 5 5. B; M 0 6, 4 (, ) Lesson 4.1 4.1 Checkpoint (pp. 17 174) 1. Because this triangle has congruent sides, it is isosceles.. Because this triangle has congruent sides, it is equilateral.. Because this triangle has no congruent sides, it is scalene. 4. Because this triangle has angles with measures less than 90 and congruent sides, it is an acute isosceles triangle. 5. Because this triangle has angles with measures less than 90 and no congruent sides, it is an acute scalene triangle. 6. Because this triangle has one angle greater than 90 and congruent sides, it is an obtuse isosceles triangle. 4.1 Guided Practice (p. 175) 1. An obtuse triangle has one angle that is greater than 90 and an acute triangle has no angles that are greater than 90.. QR **** is the side that is opposite ap.. PR **** is the side that is opposite aq. 4. Because this triangle has congruent sides, it is isosceles. 5. Because this triangle has congruent sides, it is isosceles. 6. Because this triangle has congruent sides, it is equilateral. 7. Because this triangle has no congruent sides, it is scalene. 8. Because this triangle has angles with measures less that 90, it is an acute triangle. 9. Because this triangle has a right angle, it is a right triangle. 10. Because this triangle has congruent angles, it is an equiangular triangle. 4.1 Practice and Applications (pp. 176 178) 11. scalene 1. isosceles 1. equilateral 14. equilateral 15. scalene 16. isosceles 17. obtuse 18. acute 19. right 0. acute 1. right. equiangular. An acute triangle has three acute angles, so the triangle is not an acute triangle. An obtuse triangle has one obtuse angle and two acute angles. 4. acute isosceles triangle 5. right isosceles triangle 6. obtuse isosceles triangle 7. right scalene triangle 8. obtuse isosceles triangle 9. acute scalene triangle 0. B 1. E. A. D 4. G 5. C 6. F 7. acute triangle 8. right triangle 9. acute triangle 40. A, B, and E; A, C, and D; A, D, and E; or B, C, and E 41. B, C, and E; A, D, and E; A, B, and E; or B, D, and E 4. CB **** is opposite aa; 4. EF **** is opposite ad; AC **** is opposite ab; DE **** is opposite af; AB **** is opposite ac. DF **** is opposite ae. 44. HJ **** is opposite ag; 45. LM **** is opposite ak; GH **** is opposite aj; KM **** is opposite al; GJ **** is opposite ah. KL **** is opposite am. 46. NQ **** is opposite ap; 47. ST **** is opposite ar; NP **** is opposite aq; RT **** is opposite as; PQ **** is opposite an. RS **** is opposite at. 48. Sample answer: 49. 50. Sample answer: 51. Sample answer: 1 5. Sample answer: 5. Sample answer: 4 14 15 4 5 10 78 78 Geometry, Concepts and Skills 51
4.1 Standardized Test Practice (p. 178) 54. C 55. F 4.1 Mixed Review (p. 178) 56. 4x (6x 10) 90 10x 10 90 10x 80 x 8 57. (11x 7) (5x ) 180 16x 4 180 16x 176 x 11 58. 50 (8x) 90 8x 40 x 5; 50 8(5) (y) 180 90 y 180 y 90 y 45 59. (, 5) (, 5 4) (0, 9) 60. (1, ) (1, 4) ( 1, 1) 61. ( 1, ) ( 1, 4) (, 6) 6. (0, 5) (0, 5 4) (, 1) 6. ( 4, ) ( 4, 4) ( 6, ) 64. (0, 0) (0, 0 4) (, 4) 65. ( 6, 4) ( 6, 4 4) ( 8, 8) 66. (, 1) (, 1 4) ( 5, ) 4.1 Algebra Skills (p. 178) 67. 5x 15 180 5x 195 x 9 68. x x 6 180 69. x 5x 0 180 x 6 180 8x 0 180 x 144 8x 160 x 48 x 0 70. x (x 8) 180 x x 8 180 4x 8 180 4x 188 x 47 71. (x 1) x 7 180 x x 7 180 x 5 180 x 175 x 175 7. 4(x 1) 9x 10 180 1x 4 9x 10 180 x 6 180 x 174 x 58 Lesson 4. 4. Checkpoint (pp. 180 181) 1. maa mab mac 180 maa 65 50 180 maa 115 180 maa 65. maa mab mac 180 45 mab 60 180 mab 105 180 mab 75. maa mad 90 maa 50 90 maa 40 maa mac 90 40 mac 90 mac 50 4. ma 60 60 10 5. ma 15 0 155 6. ma4 55 58 11 4. Guided Practice (p. 18) 1. 4 B A 1 5 6 C Answers may vary.. A. 70 49 x 180 4. 94 x 16 119 x 180 x 4 x 61 5. 55 x 90 x 5 5 Geometry, Concepts and Skills
4. Practice and Applications (pp. 18 184) 6. ma1 78 1 180 ma1 109 180 ma1 71 7. ma1 40 0 180 ma1 70 180 ma1 110 8. ma1 8 90 ma1 5 9. ma1 60 60 180 ma1 10 180 ma1 60 10. ma1 100 4 180 ma1 14 180 ma1 7 11. ma1 45 90 ma1 45 1. ma 98 50 148 1. ma 64 75 19 14. ma 10 145 ma 4 15. ML **** is opposite amnl. 16. 7 mal 90 mal 5 17. 0 mal 90 mal 60 45 mal 90 mal 45 The downstream angle should be between 45 and 60. 18. x x (x 15) 180 5x 15 180 5x 165 x 19. 6x 8 8 6x 10 x 0 0. x 4 90 x 48; y 100 48 180 y 148 180 y 1. 45 50 (x 5) 180 x 100 180 x 80 x 40; x y 90 40 y 90 y 50. For any position of point C, mapbc mabac mabca. This illustrates the Exterior Angle Theorem.. x x 6 180 6 x 6 180 x 154 x 77 4. x P x 6 5x 5x x 6 180 6x 6 180 6x 144 x 4 mar x 4 ; maq 5x 5(4) 10 4. Standardized Test Practice (p. 184) 5. C; (x ) 18 180 x 148 180 x x 16 x 6. F; the exterior angle measures are as follows: 50 60 110, 60 70 10, and 50 70 10 4. Mixed Review (p. 184) 7. m n by the Corresponding Angles Converse Postulate. 8. m n by the Alternate Exterior Angles Converse Theorem. 9. m n by the Same-Side Interior Angles Converse Theorem (118 6 180 ). 4. Algebra Skills (p. 184) R 0. 1015 1051 1..5.06. 8.09 8.1. 1.75 1.57 4. 0 0.5 5..055.1 Geometry, Concepts and Skills 5
Quiz 1 (p. 184) 1. obtuse isosceles triangle. acute scalene triangle. right scalene triangle 4. ma1 60 90 5. ma1 8 40 78 ma1 0 6. ma1 85 150 ma1 65 Lesson 4. 4. Geo-Activity (p. 185) Step. ah and ak are congruent. Step. For each isosceles triangle, ah and ak are congruent. 4. Checkpoint (p. 186) 1. y 50. y 9. y 4 16 y 1 4. Guided Practice (p. 188) 1. Equilateral means that all sides are congruent. Equiangular means that all angles are congruent.. LM **** MN **** NL **** ; al am an. **** ST RS **** ; ar at 4. UW **** UV **** ; aw av 5. x 50 by the Base Angles Theorem 6. x 8.8 by the Converse of the Base Angles Theorem 4. Practice and Applications (pp. 188 190) 7. x 55 by the Base Angles Theorem 8. x 68 by the Base Angles Theorem 9. x 45 by the Corollary to the Triangle Sum Theorem and the Base Angles Theorem 10. x 4 11 11. 6x 1 x 7 x 1. x 1 1. 7x 5 19 14. (5x 7) 5 7x 14 5x 45 x x 9 15. x x 4x 180 10x 180 x 18 16. By definition, an isosceles triangle is a triangle with at least congruent sides. Every equilateral triangle has congruent sides (therefore it has at least congruent sides.) So, every equilateral triangle is also isosceles. 17. x 0 0 180 18. x 50 50 180 x 60 180 x 100 180 x 10 x 80 So, maa 10. So, maa 80. 19. x x x 180 0. y 11 4x 180 x 45 So, maa (45 ) 90. 1. y 10 y 5. 4y y 5. y 5 y 8 y 4 4. y 5y 14 5. 8y 10 4y y 14 4y 1 y 7 y 6. First, show the T XYZ is equiangular and therefore max may maz 60. Then use the Corresponding Angles Postulate to show that mayjk maxjl 60, maykj malkz 60, and maxlj mazlk 60. With these measures you can use the Triangle Sum Theorem three times to show that majkl maljk majlk 60. Then you can state that T JKL is equiangular and therefore is also equilateral. 7. No, because the triangle would not be isosceles. 8. Yes, because when sides of a triangle are congruent, then the angles opposite them are congruent (Base Angles Theorem). 9. Z W X Y V 0. Because VX **** WX ****, axwv axvw by the Base Angles Theorem. 1. T WXV, T VXY, T YXZ, T ZXW. Yes, T ABC is isosceles. (Note that two sides of the triangle are radii of the circle, and all radii of a circle have the same length.) 4. Standardized Test Practice (p. 190). A; maefd maefg 180 maefd 15 180 maefd 55 54 Geometry, Concepts and Skills
4. G; madef maedf maefd 180 madef 55 55 180 madef 110 180 madef 70 4. Mixed Review (p. 190) 5. madbc maabd 4 ; maabc (maabd) (4 ) 84 6. madbc 1 (maabd) 1 (56 ) 8 ;ˆ maabc madbc 8 7. madbc (maabd) (75 ) 150 ; maabc maabd 75 8. (x 0) 55 9. (x 8) 4 x 5 x 50 40. (x 1) 81 x 80 x 40 4. Algebra Skills (p. 190) 41. 4 9 7 7 7 4. 1 1 1 1 1 1 11 4. 1 1 1 1 44. 4 0 0 0 0 0 Lesson 4.4 4.4 Activity (p. 191) 1. yes; 9 9 18. Yes, the sum of the areas from the two legs is equal to the area of the square from the hypotenuse.. There are 9 full squares contained in the figure and 4 triangles. If you combine triangles, they are 4 full squares. So, the area is (full squares) (two triangles combined) 9 (4) 17. 4. When squares are drawn from each side of a right triangle, the sum of the area of the squares from the two legs is equal to the area of the square from the hypotenuse. 4.4 Checkpoint (pp. 19 194) 1. a 6 10. b 15 17 a 6 100 b 5 89 a 64 b 64 a 6 4 8 b 6 4 8. c 7 8 c 49 64 c 11 c 1 1 10.6 4. AB ( 0 ) ( 4 0 ) 4 9 1 6 5 5 The distance between A and B is 5 units. 5. DE ( 1 ) ( 4 ) ( 6 ) 4 6 4 0 6. The distance between D and E is about 6. units. 6. FG ( ( )) ( ) ( 1 ) ( 5 ) 1 5 6 5.1 The distance between F and G is about 5.1 units. 4.4 Guided Practice (p. 195) 1. Sample answer:. x 8 10 A b C a c If mac 90, then a b c. x 64 100 x 6 x 6. x 1 4. x 4 8 x 1 4 x 16 64 x 5 x 48 x 5. x 4 8 6.9 5. AB (5 0 ) ( 0 ) 5 5 9 4 5.8 units 6. CD (4 ) ( 6 1 ) 5 B 4 5 9 5.4 units 7. FG ( 1 ) ( ( )) 6 4 6 4 0 6. Geometry, Concepts and Skills 55
4.4 Practice and Applications (pp. 195 198) 8. c 9 1 9. c 9 40 c 81 144 c 81 1600 c 5 c 1681 c 5 15 c 1 6 8 1 41 10. c 65 7 11. c 10 4 c 45 5184 c 100 576 c 9409 c 676 c 9 4 0 9 97 c 6 7 6 6 1. c 1 5 1. c 8 15 c 144 15 c 64 5 c 169 c 89 c 1 6 9 7 c 8 9 17 14. b 4 5 15. a 9 89 b 576 65 a 151 791 b 49 a 6400 b 7 a 80 16. b 5 6 1 b 5 61 b 6 b 6 17. b 5 18. c 4 6 b 9 5 c 16 6 b 16 c 5 b 4; c 5 7.; The side lengths form The side lengths do not a Pythagorean Triple. form a Pythagorean Triple. 19. c 7 11 c 49 11 c 170 c 1 7 0 1.04; The side lengths do not form a Pythagorean Triple. 0. c 16 0 c 56 900 c 1156 c 1 1 5 6 4; The side lengths form a Pythagorean Triple. 1. b 9 4 b 81 576 b 495 b 4 9 5.; The side lengths do not form a Pythagorean Triple.. a 48 50 a 04 500 a 196 a 1 9 6 14; The side lengths form a Pythagorean Triple.. Yes, because 0 1 400 441 841 9 ; 4. Yes, because 7 4 49 576 65 5 ; 5. No, because 5 1 5 144 169 14. 6. x.6 47.57 x 541.076 6.9049 x 80.95 x 8 0.9 5 5.95 Each support beam is approximately 5.95 m. 7. AB (5 ) ( ( )) 4 9 1 6 5 5 units 8. CD (6 0 ) ( 8 ) 6 6 6 6 7 8.5 units 9. EF (5 4 ) ( 5 ( 1 )) 1 6 0. 0 7 1 6 7 6.1 units y 1 9 5 4 1 1 R( 1, ) (1, 6) PQ (4 1 ) ( 4 ( 6 )) 9 4 1.6 QR ( 1 1 ) ( ( 6 )) ( ) 4 9 1.6 So, PQ **** QR ****. x P(4, 4) 56 Geometry, Concepts and Skills
1.. ( 8, 5) PQ ( 8 ( 1 )) ( 5 ( 6 )) ( 7 ) 1 1 4 9 1 1 1 7 0 1.04 QR ( 8 ) ( 5 ( )) ( 1 1 ) 7 1 1 4 9 1 7 0 1.04 So, PQ **** QR ****. R(, 6) P( 1, 6) y y R(, ) P(5, 1) ( 5, 7) PQ (5 ( 5) ) ( 1 ( 7) 1 0 8 1 0 0 6 4 1 6 4 1.81 QR ( 5 ( )) ( 7 6 ) ( ) ( 1 ) 4 1 6 9 1 7 1. So, PQ **** is not congruent to QR ****.. A to B AD DE EB (1 0 0 5 0 ) ( 0 0 ) (1 0 0 1 0 0 ) ( 0 0 ) (1 0 0 6 5 ) ( 0 0 ) 5 0 ( 0 ) 5 50 0 5 115 yd; B to C BF FC (6 5 0 ) ( 0 0 ) (0 0 ) ( 0 1 5 ) 6 5 1 5 65 15 80 yd; C to A CG GA (0 0 ) ( 1 5 0 ) (0 5 0 ) ( 0 0 ) 1 5 ( 5 0 ) 15 50 65 yd x x 4. A to B (6 5 5 0 ) ( 0 0 ) 1 5 0 ( 5 9 0 0 ) 1 1 5.5 yd; B to C (6 5 0 ) ( 0 1 5 ) 6 5 1 5 4 5 5 4 4 5 0 66.7 yd; C to A (0 5 0 ) ( 1 5 0 ) 5. x 8 16 5 0 1 5 5 0 0 5 7 5 5. yd x 64 56 x 19 x 1 9 1.9 6. y 6 10 7. y 5 y 6 100 y 9 5 y 64 y 16 y 6 4 8 y 1 6 4 1 8 4 11 4 7 x 4 6 x 7 x 16 6 x 49 9 x 5 x 58 x 5 7. x 5 8 7.6 4.4 Standardized Test Practice (p. 198) 8. D; d ( 1 ) ( 4 5 ) ( 4 ) ( 9 ) 1 6 8 1 9 7 9. F; c 56 1089 16 45 c 4 5 65 4.4 Mixed Review (p. 198) 40. 7 41. 1.05 4. 0 4. 0.0 44. x 5 8 45. 10 x 1 x x 46. 4x 8 x 7 47. 6x 11 11 6x 0 x 0 Geometry, Concepts and Skills 57
4.4 Algebra Skills (p. 198) 48. 5 49. 5 50. 7 50 51. 5 5. 1 4 5. 17 54. 1 1000 55. 1 9 Quiz (p. 198) 1. x 5 1. (x 1) 55 x 18 x 54 x 6 x 7. x 5 4x 16 x 1 x 7 4. AB ( ) ( 0 ) ( 5 ) 5 9 4 5.8 units 5. AB ( 1 ) ( 1 ( )) ( 4 ) 1 1 6 1 1 7 4.1 units 6. AB ( 1 1 ) ( 1 ) ( ) ( ) 4 9 1.6 units 7. d 5 6 d 5 6 d 11 d 1 1. ft Lesson 4.5 4.5 Technology Activity (p. 199) 1. maacb 90. maacb 90. maacb 90 4. no 4.5 Checkpoint (p. 0) 1. 6 5. 17 8 15 6 5 4 89 64 5 6 9; 89 89; The triangle is obtuse. The triangle is right.. 7 7 7 4. 4 4 7 49 49 49 576 576 49 49 98; 576 65 The triangle is acute. The triangle is acute. 5. 5 4 7 6. 6 4 7 65 576 49 676 576 49 65 65 676 65 The triangle is right. The triangle is obtuse. 4.5 Guided Practice (p. 0) 1. Sample answer: If you square the lengths of the two shortest sides of a triangle and add the results, and the sum is equal to the square of the length of the longest side of the triangle, then the triangle is a right triangle.. 5 4 4. 14 6 1 5 16 16 196 6 144 5 196 180 The triangle is acute. The triangle is obtuse. 4. 15 1 9 5 144 81 5 5 The triangle is right. 5. C; 11 10 6. B; 8 5 7 11 4 100 64 5 49 11 104 64 74 7. D; The side lengths are equal so the triangle is equiangular. 8. A; 10 6 8 100 6 64 100 100 4.5 Practice and Applications (pp. 0 05) 9. 5 0 15 10. 6 10 4 65 400 5 676 100 576 65 65 676 676 The triangle is right. The triangle is right. 11. 1 7 1 4 1. 1 11 17 1 16 59 441 11 17 17 59 56 The triangle is right. The triangle is acute. 1. 7 6 4 14. 9 5 49 6 16 9 5 9 49 5 9 4 The triangle is acute. The triangle is acute. 15. 7 6 16. 1 8 6 49 4 6 144 64 6 49 40 144 100 The triangle is obtuse. The triangle is obtuse. 58 Geometry, Concepts and Skills
17. 16 1 18. 1 0 484 56 169 59 144 400 484 45 59 544 The triangle is obtuse. The triangle is acute. 19. 1 16 9 0. 9 6 15 441 56 81 151 196 5 441 7 151 151 The triangle is obtuse. The triangle is right. 1. 6 18 17. 5 0 48 676 4 89 704 400 04 676 61 704 704 The triangle is obtuse. The triangle is right.. 16 14 10 56 196 100 56 96 The triangle is acute. 4. 10 119 169 14,400 14,161 8,561 8,561 8,561; 4800 4601 6649,040,000 1,169,01 44,09,01 44,09,01 44,09,01; 1,500 1,709 18,541 18,50,000 161,518,681 4,768,681 4,768,681 4,768,681 5. 101 0 99 6. 5 1 8 10,01 400 9801 15 441 784 10,01 10,01 15 15 The triangle is right. The triangle is right. 7. 6 10 17 8. 11 7 10 676 100 89 11 49 100 676 89 11 149 The triangle is obtuse. The triangle is acute. 9. 9 6 7 4 0. 7 1 6 81 67 16 49 1 6 81 8 49 49 The triangle is acute. The triangle is right. 1. 757 468 595 57,049 19,04 54,05 57,049 57,049 The triangle is right.. 14 10 11. 5 4 5 196 100 11 5 16 5 196 1 5 41 The triangle is acute. The triangle is acute. 4. 145 17 144 5. 50 10 49 1,05 89 0,76 500 100 401 1,05 1,05 500 501 The triangle is right. The triangle is acute. 6. 5.5 5 5 0.5 5 5 0.5 0 The triangle is obtuse. 7. 714 40 599 509.796 16,409 58,801 509,796 51,10 The triangle formed between the cities is not a right triangle. 8. No; For Tallahassee to be directly south of Cincinnati, the triangle formed by the three cities would have to be a right triangle. As shown in Exercise 7, the triangle is not a right triangle. 9. No; Sample answer: A counterexample is the right triangle with side lengths, 4, and 5. Double each side to get 6, 8, and 10. 10 6 8 40. 100 6 64 100 100 The side lengths 6, 8, and 10 form a right triangle, not an obtuse triangle. P(, 4) R( 6, ) PQ ( 5 ) ( 4 0 ) ( 8 ) 4 6 4 1 6 8 0 QR (5 ( 6) ) ( 0 ( ) ) 1 1 1 1 4 1 5 RP ( ( 6 )) ( 4 ( )) 6 9 6 4 5 1 5 8 0 4 5 15 80 45 15 15 So, T PQR is right. y 4 (5, 0) x Geometry, Concepts and Skills 59
41. P( 1, ) 1 PQ ( 1 4 ) ( 1 ) ( 5 ) 1 5 1 6 QR (4 0 ) ( 1 ( 1) ) 4 1 6 4 0 RP (0 ( 1 )) ( 1 ) 1 ( ) 1 9 1 0 6 0 1 0 6 0 10 6 0 So, T PQR is acute. 4.5 Standardized Test Practice (p. 05) 4. a. 4 90 100 b. 4 90 99 1764 8100 10,000 1764 8100 9801 9864 10,000 9864 9801 T ABC is obtuse. T ABC is acute. c. Sample answer: Choose AB to be 4 feet. AC 4 90 1764 8100 9864 AC 9 8 6 4 99. feet If AB 4 feet and AC 99. feet, then T ABC would be a right triangle. 4.5 Mixed Review (p. 05) 4. x 67 1 y R(0, 1) (4, 1) x 44. x 4 8 45. 5x 5x x 180 x 1 1x 180 x 4 x 15 4.5 Algebra Skills (p. 05) 46. 5 47. 9 48. 1 4 49. 1 50. 4 1 51. 14 5. 1 6 7 5. 1 Lesson 4.6 4.6 Activity (p. 06) 1. Answers will vary, but in each column the second and third entries should be approximately equal.. Yes; the distance from P to a vertex is equal to two thirds the distance from that vertex to the midpoint of the opposite side.. The results are the same for any triangle. 4.6 Checkpoint (pp. 07 09) 1. Sample answer:. Sample answer:. Sample answer: 5 4. BE BD (4) 16; BE ED BD 16 ED 4 ED 8 5. KG JG KG JK JG JG 4 JG 1 JG 4 JG 1; KG (1) 8 6. QN PN 0 PN 5 (0) PN 0 PN; PQ QN PN PQ 0 0 PQ 10 5.5.5 5 60 Geometry, Concepts and Skills
4.6 Guided Practice (p. 09) 1. The segment from a vertex of a triangle to the midpoint of the opposite side is a median.. The centroid is the point where the three medians intersect.. A 9 B 4. AD 4 5. AD 11 6. AD 7. PT PS () ; PT ST PS ST ST 11 8 5 5 C 8. BE BD 1 BD (1) BD 18 BD; BE ED BD 1 ED 18 ED 6 14. CD CE 15. CD CE CD DE CE CD DE CE CE 5 CE CE 11 CE 5 1 CE 11 1 CE (5) CE (11) CE 15 CE; CE; CD DE CE CD DE CE CD 5 15 CD 11 CD 10 CD 16. CD CE CD DE CE CE 9 CE 9 1 CE (9) CE 7 CE; CD DE CE CD 9 7 CD 18 17. No. The median starts at aa and goes to BC ****. The segment that starts at aa and goes to the centroid D is the one that is the median. So, AD AE (18) 1 and 4.6 Practice and Applications (pp. 10 11) 9. F 10. E 11. PN QN 1. PN QN (9) 6; (1) 14; PN QP QN PN QP QN 6 QP 9 14 QP 1 QP QP 7 1. PN QN (0) 0; PN QP QN 0 QP 0 QP 10 G J K L AD DE AE 1 DE 18 DE 6. 18. Sample answer: P 19. Q 1 11, (5, 0) R 1 5, 6 (, ) S 5 11, 6 (8, 4) 0. MS ( 1 8 ) ( 4 ) ( 9 ) ( 6 ) 8 1 6 1 1 7 10.8 units NR (1 1 ) ( ) 9 0 8 1 9 units PQ (5 5 ) ( 6 0 ) 0 6 6 6 units Geometry, Concepts and Skills 61
1. PT PQ (6) 4 Because PQ is vertical, T is 4 units directly below P. P(5, 6) so T(5, 6 4) (5, ) 4.6 Standardized Test Practice (p. 11). B; KN KM (6) 4; KN NM KM 4 NM 6 NM 1. G; VT PT VT 1 VT (1) 18 4.6 Mixed Review (p. 11) 4. ma1 45 5 180 ma1 70 180 ma1 110 5. ma1 ma1 60 180 (ma1) 60 180 (ma1) 10 ma1 60 6. ma1 5 90 ma1 55 7. ma1 60 65 15 8. ma1 40 4 8 9. ma1 115 147 4.6 Algebra Skills (p. 11) In Exercises 0 41, sample answers are given. 0. 4, 5 10 1. 15, 1 0 50. 1 16, 1 8. 4 7, 4 8 84 4. 4 5, 1 0 1 5. 60 1 0, 6 60 6. 9, 4 18 7. 9, 0 8. 15 5, 56 140 9. 4, 90 40. 10, 00 8 41. 00 1 1, 4 8 66 Lesson 4.7 4.7 Checkpoint (pp. 1 14) 1. LM MN, so man mal. MN NL, so mal mam. The order of the angles from largest to smallest is an, al, am.. PR PQ, so maq mar. PQ QR, so mar map. The order of the angles from largest to smallest is aq, ar, ap.. ST TU, so mau mas. TU US, so mas mat. The order of the angles from largest to smallest is au, as, at. 4. maj mah, so HG GJ. mah mag, so GJ JH. The order of the sides from longest to shortest is HG ****, GJ ****, JH ****. 5. maf mad, so DE EF. mad mae, so EF FD. The order of the sides from longest to shortest is DE ****, EF ****, FD ****. 6. mab mac, so AC BA. mac maa, so BA CB. The order of the sides from longest to shortest is AC ****, BA ****, CB ****. 7. The side lengths 5, 7, 1 can not form a triangle because 5 7 1. 8. The side lengths 6, 9, 1 can form a triangle because 6 9 1, 6 1 9, and 9 1 6. 9. The side lengths 10, 15, 5 can not form a triangle because 10 15 5. 4.7 Guided Practice (p. 14) 1. The symbol means greater than, and the symbol means less than.. aa is opposite the shortest side so it is the smallest angle.. AC **** is opposite the largest angle so it is the longest side. 4. mad mae maf 180 mae 10 180 mae 15 180 mae 45 ad is the smallest angle of T DEF. af is the largest angle of T DEF. 5. EF **** is the shortest side of T DEF. DE **** is the longest side of T DEF. 6. 1,, can not form a triangle because 1. 7. 6, 10, 15 can form a triangle because 6 10 15, 6 15 10, and 10 15 6. 8. 1, 16, 0 can not form a triangle because 1 16 0. 9. 7, 8, 1 can form a triangle because 7 8 1, 7 1 8, and 8 1 7. 6 Geometry, Concepts and Skills
10. 4, 9, 16 can not form a triangle because 4 9 16. 11. 5, 5, 10 can not form a triangle because 5 5 10. 4.7 Practice and Applications (pp. 15 18) 1. ac is the smallest angle of T ABC; a B is the largest angle of T ABC. 1. ar is the smallest angle of T PQR; aq is the largest angle of T PQR. 14. ah is the smallest angle of T GHF; af is the largest angle of T GHF. 15. RT **** is the shortest side of T RST; **** TS is the longest side of T RST. 16. maa mab mac 180 maa 4 71 180 maa 11 180 maa 67 AC **** is the shortest side of T ABC. BA **** is the longest side of T ABC. 17. mah maj 90 5 maj 90 maj 55 KJ **** is the shortest side of T JKH. JH **** is the longest side of T JKH. 18. LK LM, so mam mak. LM MK, so mak mal. am, ak, al 19. NQ PN, so map maq. PN PQ, so maq man. ap, aq, an 0. TS TR, so mar mas. TR RS, so mas mat. ar, as, at 1. AB BC, so mac maa. BC AC, so maa mab. ac, aa, ab. XW YW, so may max. YW XY, so max maw. ay, ax, aw. EF DF, so mad mae. DF DE, so mae maf. ad, ae, af 4. Since 8 is the largest angle; the side opposite it (from the sink to the refrigerator) should be the longest. But the labels show that the line from the refrigerator to the stove is longer. 5. No. A kitchen triangle could not have the side lengths of 9 ft, ft, and 5 ft because 5 9. 6. mab maa, so AC BC. maa mac, so BC AB. AC ****, CB ****, BA **** 7. mae maf 90 0 maf 90 maf 60 mad maf, so EF DE. maf mae, so DE DF. EF ****, DE ****, FD **** 8. mah mag maj 180 5 mag 10 180 mag 155 180 mag 5 maj mah, so HG GJ. mah mag, so GJ JH. HG ****, GJ ****, JH **** 9. maa mab mac 180 44 95 mac 180 19 mac 180 mac 41 mab maa, so AC BC. maa mac, so BC AB. AC ****, CB ****, BA **** 0. mar maq 90 mar 50 90 mar 40 map maq, so QR PR. maq mar, so PR PQ. RQ ****, PR ****, QP **** 1. maf mag mah 180 maf 58 6 180 maf 10 180 maf 60 mah maf, so FG GH. maf mag, so GH FH. GF ****, HG ****, FH ****. The side lengths do not satisfy the triangle inequality because 5. Geometry, Concepts and Skills 6
. The side lengths do not satisfy the Triangle Inequality because 10 14. 4. 5. 10, 1, can not form a triangle because 10 1. 6. 17, 9, 0 can not form a triangle because 17 9 0. 7. 8. A 9.7 cm 9. Cutting across the empty lot is shorter than taking the sidewalks because the path through the lot is the hypotenuse of a right triangle and the sum of the two legs must be greater than the hypotenuse. 40. No, your friend can not be right because 1.8 1.5 4.6. By the Triangle Inequality, you must be more than 4.6 1.8.8 miles from camp. 41. When the boom lines are shortened, the boom is raised. 4. 150 4 cm 59 10 cm 75 1 cm 5 cm 15 B 11.5 cm 46 7.1 cm 7 cm 15.6 cm 0 4. Yes, aacb can be larger than abac; the maximum length for AB **** is 150 ft. Since BC is only 100 ft, when AB BC, the angle opposite AB **** (aacb) would be larger than the angle opposite BC **** (abac). 4.7 Standardized Test Practice (p. 17) C d. Sample answer: 1 in., 7 in., 10 in.; in., 5 in., 10 in.; in., 7 in., 9 in. 4.7 Mixed Review (p. 18) 45. RT **** is the hypotenuse of T RST. 46. In T RST, RT **** is the side opposite arst. 47. The legs of T RST are RS **** and ****. ST 48. RT **** is the base of T RST. 49. ma1 79 4 180 ma1 1 180 ma1 58 50. ma 8 5 180 ma 5 180 ma 17 51. ma 56 90 ma 4 4.7 Algebra Skills (p. 18) 5. x 6 1 5 5 5. 1 8 6 x 15x 0 18x 18 x x 1 x 6 54. 7 7 55. 7 1 x 7x 4 7x 189 x 6 x 7 56. 5 8 x 7 7 10 9 x 8x 60 7x 490 x 45 x 70 44. a. 6 in. 6 in. 5 in. 5 in. 8 in. 6 in. 8 in. 8 in. 7 in. 7 in. 4 in. in. b. c. 6 in. 5 in. 7 in. 8 in. in. 7 in. Quiz (p. 18) 1. 14 6 11. 16 15 7 196 6 11 56 5 49 196 157 56 74 The triangle is obtuse. The triangle is acute.. 8 18 80 674 4 6400 674 674 The triangle is right. 4. KN KM 5. KN KM (6) 4; (9) 6; KN MN KM KN MN KM 4 MN 6 6 MN 9 MN MN 1 64 Geometry, Concepts and Skills
6. KN KM (60) 40; KN MN KM 40 MN 60 MN 0 7. mal maq, so MQ LM. maq mam, so LM QL. MQ ****, LM ****, QL **** 8. mam maq, so QP PM. maq map, so PM MQ. QP ****, PM ****, MQ **** 9. map mam, so MN NP. mam man, so NP PM. MN ****, NP ****, PM **** Chapter 4 Summary and Review (pp. 19 ) 1. A triangle is a figure formed by three segments joining three noncollinear points.. The side opposite the right angle is the hypotenuse of a right triangle.. A corollary to a theorem is a statement that can be proved easily using the theorem. 4. The congruent sides of an isosceles triangle are called legs, and the third side is called the base. 5. A point that joins two sides of a triangle is called a vertex. 6. A segment from a vertex of a triangle to the midpoint of its opposite side is called a median. 7. The point at which the medians of a triangle intersect is called the centroid of a triangle. 8. isosceles 9. equilateral 10. scalene 11. right 1. acute 1. isosceles 14. ma1 5 6 115 15. ma1 10 40 14 16. ma1 8 54 17 17. 16 x x 180 x 16 180 x 164 x 8 The other interior angles each measure 8. 18. 1 x 90 x 59 One of the interior angles is 90 and the other is 59. 19. x 8 1 x 5 0. (x 5) 49 x 44 x 1. 4x 16 x 4. x x 10. 64 64 x 180 x 10 x 18 180 x 60 x 5 4. 4x 5x 4 4 x 5. x 40 0 6. x 10 x 1600 900 x 100 484 x 500 x 84 x 5 0 0 50 x 8 4 19.6 7. x 6 16 x 6 56 x 0 x 0 14.8 8. AB ( 0 ) ( 4 0 ) ( ) 4 9 1 6 5 5 9. AB (6 ) ( 4 5 ) 4 ( 9 ) 1 6 8 1 9 7 9.8 0. AB ( ( 8 )) ( 7 7 ) 1 1 0 1 1 11 1. AB (0 ( 4 )) ( 6 ( 1 )) 4 7 1 6 4 9 6 5 8.1. AB ( 6 ( )) ( 7 ( 1 )) ( 4 ) ( 6 ) 1 6 6 5 7.. AB ( 8 ) ( 4 ( )) ( 1 0 ) 7 1 0 0 4 9 1 4 9 1. Geometry, Concepts and Skills 65
4. AB ( 9 ) ( 6 1 ) ( 1 ) ( 7 ) 1 4 4 4 9 1 9 1.9 5. AB (0 5 ) ( 6 4 ) ( 5 ) 5 4 9 5.4 6. 15 1 9 7. 16 7 11 5 144 81 56 49 11 5 5 56 170 right obtuse 8. 18 19 9. 44 18 4 484 4 61 196 4 1764 484 685 196 088 acute acute 40. 1 10 41. 1 15 1 144 100 9 961 5 441 144 109 961 666 obtuse obtuse 4. KP KM 4. KP KM (18) 1; (4) 8; KP PM KM KP PM KM 1 PM 18 8 PM 4 PM 6 PM 14 44. KP KM (10) 80; KP PM KM 80 PM 10 PM 40 45. CD CE 46. CD CE 8 CE 16 CE (8) CE (16) CE 1 CE; 4 CE; CD DE CE CD DE CE 8 DE 1 16 DE 4 DE 4 DE 8 47. CD CE 8 CE (8) CE 4 CE; CD DE CE 8 DE 4 DE 14 48. QR PR, so map maq. PR PQ, so maq mar. ap, aq, ar 49. TS US, so mau mat. US UT, so mat mas. au, at, as 50. XZ YZ, so may max. YZ XY, so max maz. ay, ax, az 51. mab maa, so AC CB. maa mac, so CB BA. AC ****, CB ****, BA **** 5. mae mad, so DF FE. mad maf, so FE ED. DF ****, FE ****, ED **** 5. maj mah, so GH JG. mah mag, so JG HJ. GH ****, JG ****, HJ **** 54. Yes, the side lengths can form a triangle because 10 11 0, 10 0 11, and 11 0 10. 55. Yes, the side lengths can form a triangle because 1 5, 1 5, and 5 1. 56. No, the side lengths can not form a triangle because 10 15. 57. No, the side lengths can not form a triangle because 6 6 1. 58. Yes, the side lengths can form a triangle because 1 14 15, 1 15 14, and 14 15 1. 59. Yes, the side lengths can form a triangle because 4, 4, and 4. 60. No, the side lengths can not form a triangle because 4 5 9. 61. Yes, the side lengths can form a triangle because 11 11 0 and 11 0 11. 6. No, the side lengths can not form a triangle because 14 0 8. 66 Geometry, Concepts and Skills
Chapter 4 Test (p. 4) 1. T JLM. T JKL. T JKL 4. T JLM 5. x (x ) (x 15) 180 6x 18 180 6x 198 x 6. 1x 54 78 7. x 15 1x 1 x 5 x 11 8. x 10 14 9. 10. x 100 196 x 96 x 9 6 17. PQ ( 6 0 ) ( 8 0 ) ( 6 ) ( 8 ) 6 6 4 1 0 0 10 (, 6) y P(0, 0) x ( 6, 8) 1 PQ ( ( )) ( 4 6 ) 4 ( ) 1 6 4 0 4.5 11. 4 17 18 1. 1 6 9 576 89 4 169 6 81 576 61 169 117 acute obtuse 1. 0 1 14 400 144 196 400 40 obtuse y 1 P(, 4) x 14. DC EC () ; DC DE EC DE DE 11 15. mab maa, so AC BC. maa mac, so BC AB. AC ****, CB ****, BA **** 16. ma 44 90 ma 46 ; ma 1 ; 1 ma ma1 180 1 1 ma1 180 6 ma1 180 ma1 118 17. No, the side lengths can not form a triangle because 5 8 18. 18. Yes, the side lengths can form a triangle because 0 4 40, 0 40 4, and 4 40 0. 19. No, the side lengths can not form a triangle because 7 7 14. 0. Yes, the side lengths can form a triangle because 1 45 50, 1 50 45, and 45 50 1. Chapter 4 Standardized Test (p. 5) 1. D. H; mabcd 90 5 15. A; 14 8 4. F 5. C; JK (8 ) ( 5 ) 5 ( 7 ) 5 4 9 7 4 6. G; 8 7 4 7. B; x 4 64 49 16 x 7 64 65 acute 8. x y 90 9. 10 8, so x y. 10. G Chapter 4 Algebra Review (p. 7) 9 hours 9 1. 4 hours 4 8 Geometry, Concepts and Skills 67
. 10 inches 10 inches feet 1 inches 1 0 inches 1 0 4 inches 4 1 5. 40 minutes 40 minutes 4 hours 4 60 minutes 40 minutes 40 40 1 40 minutes 40 40 6 6 pounds 4. 6 16 ounces 0 ounces 0 ounces 9 6 ounces 9 6 4 0 ounces 0 4 54 5. 5 6 ho 8 da urs ys 5 6 hours 8 days 8 8 7 hours 1 day 7 hours/day 6. 6 0 mil hou es rs 6 0 mile hour s s 0 miles 1 hour 0 miles/hour $ 8 $ 8 4 $9. 50 7. 4 hours 4 hours 4 1 hour $9.50/hour $ 5.88 $ 5.88 1 $. 49 8. 1 bagels 1 bagels 1 1 bagel $.49/bagel 68 Geometry, Concepts and Skills