CHAPTER 4. Chapter Opener PQ (3, 3) Lesson 4.1

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1 CHAPTER 4 Chapter Opener Chapter Readiness Quiz (p. 17) 1. D. H; PQ **** is horizontal, so subtract the x-coordinates. PQ B; M 0 6, 4 (, ) Lesson Checkpoint (pp ) 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 ) 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: Sample answer: 51. Sample answer: 1 5. Sample answer: 5. Sample answer: Geometry, Concepts and Skills 51

2 4.1 Standardized Test Practice (p. 178) 54. C 55. F 4.1 Mixed Review (p. 178) 56. 4x (6x 10) 90 10x x 80 x (11x 7) (5x ) x x 176 x (8x) 90 8x 40 x 5; 50 8(5) (y) y 180 y 90 y (, 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 x 195 x x x x 5x x x x 144 8x 160 x 48 x x (x 8) 180 x x x x 188 x (x 1) x x x x x 175 x (x 1) 9x x 4 9x x x 174 x 58 Lesson Checkpoint (pp ) 1. maa mab mac 180 maa maa maa 65. maa mab mac mab mab mab 75. maa mad 90 maa maa 40 maa mac mac 90 mac ma ma ma Guided Practice (p. 18) 1. 4 B A C Answers may vary.. A x x x 180 x 4 x x 90 x 5 5 Geometry, Concepts and Skills

3 4. Practice and Applications (pp ) 6. ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ML **** is opposite amnl mal 90 mal mal 90 mal mal 90 mal 45 The downstream angle should be between 45 and x x (x 15) 180 5x x 165 x 19. 6x 8 8 6x 10 x 0 0. x 4 90 x 48; y y y (x 5) 180 x x 80 x 40; x y y 90 y 50. For any position of point C, mapbc mabac mabca. This illustrates the Exterior Angle Theorem.. x x x x 154 x x P x 6 5x 5x x x x 144 x 4 mar x 4 ; maq 5x 5(4) Standardized Test Practice (p. 184) 5. C; (x ) x x x 16 x 6. F; the exterior angle measures are as follows: , , and 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 ( ). 4. Algebra Skills (p. 184) R Geometry, Concepts and Skills 5

4 Quiz 1 (p. 184) 1. obtuse isosceles triangle. acute scalene triangle. right scalene triangle 4. ma ma ma ma ma1 65 Lesson 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 ) 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 x 1 x 7 x 1. x x (5x 7) 5 7x 14 5x 45 x x x x 4x x 180 x 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 x x x x 10 x 80 So, maa 10. So, maa x x x y 11 4x 180 x 45 So, maa (45 ) y 10 y 5. 4y y 5. y 5 y 8 y 4 4. y 5y y 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 maefd Geometry, Concepts and Skills

5 4. G; madef maedf maefd 180 madef madef madef Mixed Review (p. 190) 5. madbc maabd 4 ; maabc (maabd) (4 ) madbc 1 (maabd) 1 (56 ) 8 ;ˆ maabc madbc 8 7. madbc (maabd) (75 ) 150 ; maabc maabd (x 0) (x 8) 4 x 5 x (x 1) 81 x 80 x Algebra Skills (p. 190) Lesson Activity (p. 191) 1. yes; 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) 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 ) 1. a b a b 5 89 a 64 b 64 a b c 7 8 c c 11 c AB ( 0 ) ( 4 0 ) The distance between A and B is 5 units. 5. DE ( 1 ) ( 4 ) ( 6 ) The distance between D and E is about 6. units. 6. FG ( ( )) ( ) ( 1 ) ( 5 ) 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 x 6 x 6. x 1 4. x 4 8 x 1 4 x x 5 x 48 x 5. x AB (5 0 ) ( 0 ) units 6. CD (4 ) ( 6 1 ) 5 B units 7. FG ( 1 ) ( ( )) Geometry, Concepts and Skills 55

6 4.4 Practice and Applications (pp ) 8. c c 9 40 c c c 5 c 1681 c 5 15 c c c 10 4 c c c 9409 c 676 c c c c 8 15 c c 64 5 c 169 c 89 c c b a 9 89 b a b 49 a 6400 b 7 a b b 5 61 b 6 b b 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 c 170 c ; The side lengths do not form a Pythagorean Triple. 0. c 16 0 c c 1156 c ; The side lengths form a Pythagorean Triple. 1. b 9 4 b b 495 b ; The side lengths do not form a Pythagorean Triple.. a a a 196 a ; The side lengths form a Pythagorean Triple.. Yes, because ; 4. Yes, because ; 5. No, because x x x x Each support beam is approximately 5.95 m. 7. AB (5 ) ( ( )) units 8. CD (6 0 ) ( 8 ) units 9. EF (5 4 ) ( 5 ( 1 )) units y R( 1, ) (1, 6) PQ (4 1 ) ( 4 ( 6 )) QR ( 1 1 ) ( ( 6 )) ( ) So, PQ **** QR ****. x P(4, 4) 56 Geometry, Concepts and Skills

7 1.. ( 8, 5) PQ ( 8 ( 1 )) ( 5 ( 6 )) ( 7 ) QR ( 8 ) ( 5 ( )) ( 1 1 ) So, PQ **** QR ****. R(, 6) P( 1, 6) y y R(, ) P(5, 1) ( 5, 7) PQ (5 ( 5) ) ( 1 ( 7) QR ( 5 ( )) ( 7 6 ) ( ) ( 1 ) So, PQ **** is not congruent to QR ****.. A to B AD DE EB ( ) ( 0 0 ) ( ) ( 0 0 ) ( ) ( 0 0 ) 5 0 ( 0 ) yd; B to C BF FC (6 5 0 ) ( 0 0 ) (0 0 ) ( ) yd; C to A CG GA (0 0 ) ( ) (0 5 0 ) ( 0 0 ) 1 5 ( 5 0 ) yd x x 4. A to B ( ) ( 0 0 ) ( ) yd; B to C (6 5 0 ) ( ) yd; C to A (0 5 0 ) ( ) 5. x yd x x 19 x y y 5 y y 9 5 y 64 y 16 y y x 4 6 x 7 x 16 6 x 49 9 x 5 x 58 x 5 7. x Standardized Test Practice (p. 198) 8. D; d ( 1 ) ( 4 5 ) ( 4 ) ( 9 ) F; c c Mixed Review (p. 198) x x 1 x x 46. 4x 8 x x x 0 x 0 Geometry, Concepts and Skills 57

8 4.4 Algebra Skills (p. 198) 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 ) units 5. AB ( 1 ) ( 1 ( )) ( 4 ) units 6. AB ( 1 1 ) ( 1 ) ( ) ( ) units 7. d 5 6 d 5 6 d 11 d 1 1. ft Lesson Technology Activity (p. 199) 1. maacb 90. maacb 90. maacb no 4.5 Checkpoint (p. 0) ; 89 89; The triangle is obtuse. The triangle is right ; The triangle is acute. The triangle is acute 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 The triangle is acute. The triangle is obtuse The triangle is right. 5. C; B; D; The side lengths are equal so the triangle is equiangular. 8. A; Practice and Applications (pp. 0 05) The triangle is right. The triangle is right The triangle is right. The triangle is acute The triangle is acute. The triangle is acute The triangle is obtuse. The triangle is obtuse. 58 Geometry, Concepts and Skills

9 The triangle is obtuse. The triangle is acute The triangle is obtuse. The triangle is right The triangle is obtuse. The triangle is right The triangle is acute ,400 14,161 8,561 8,561 8,561; ,040,000 1,169,01 44,09,01 44,09,01 44,09,01; 1,500 1,709 18,541 18,50, ,518,681 4,768,681 4,768,681 4,768, , ,01 10, The triangle is right. The triangle is right The triangle is obtuse. The triangle is acute The triangle is acute. The triangle is right ,049 19,04 54,05 57,049 57,049 The triangle is right The triangle is acute. The triangle is acute , , ,05 1, The triangle is right. The triangle is acute The triangle is obtuse ,409 58, ,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 The side lengths 6, 8, and 10 form a right triangle, not an obtuse triangle. P(, 4) R( 6, ) PQ ( 5 ) ( 4 0 ) ( 8 ) QR (5 ( 6) ) ( 0 ( ) ) RP ( ( 6 )) ( 4 ( )) So, T PQR is right. y 4 (5, 0) x Geometry, Concepts and Skills 59

10 41. P( 1, ) 1 PQ ( 1 4 ) ( 1 ) ( 5 ) QR (4 0 ) ( 1 ( 1) ) RP (0 ( 1 )) ( 1 ) 1 ( ) So, T PQR is acute. 4.5 Standardized Test Practice (p. 05) 4. a b , , T ABC is obtuse. T ABC is acute. c. Sample answer: Choose AB to be 4 feet. AC AC 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 x 5x x 180 x 1 1x 180 x 4 x Algebra Skills (p. 05) Lesson 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 ) 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 Geometry, Concepts and Skills

11 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 AD 7. PT PS () ; PT ST PS ST ST C 8. BE BD 1 BD (1) BD 18 BD; BE ED BD 1 ED 18 ED 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 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 ) 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 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 ) units NR (1 1 ) ( ) units PQ (5 5 ) ( 6 0 ) units Geometry, Concepts and Skills 61

12 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) Mixed Review (p. 11) 4. ma ma ma ma1 ma (ma1) (ma1) 10 ma ma ma ma ma ma Algebra Skills (p. 11) In Exercises 0 41, sample answers are given. 0. 4, , , , , , , , , , , , Lesson 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 The side lengths 6, 9, 1 can form a triangle because 6 9 1, 6 1 9, and The side lengths 10, 15, 5 can not form a triangle because 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 mae 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 , 10, 15 can form a triangle because , , and , 16, 0 can not form a triangle because , 8, 1 can form a triangle because 7 8 1, 7 1 8, and Geometry, Concepts and Skills

13 10. 4, 9, 16 can not form a triangle because , 5, 10 can not form a triangle because Practice and Applications (pp ) 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 maa 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 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 mag mag mag 5 maj mah, so HG GJ. mah mag, so GJ JH. HG ****, GJ ****, JH **** 9. maa mab mac mac mac 180 mac 41 mab maa, so AC BC. maa mac, so BC AB. AC ****, CB ****, BA **** 0. mar maq 90 mar mar 40 map maq, so QR PR. maq mar, so PR PQ. RQ ****, PR ****, QP **** 1. maf mag mah 180 maf maf 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

14 . The side lengths do not satisfy the Triangle Inequality because , 1, can not form a triangle because , 9, 0 can not form a triangle because 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 By the Triangle Inequality, you must be more than miles from camp. 41. When the boom lines are shortened, the boom is raised cm cm 75 1 cm 5 cm 15 B 11.5 cm 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. ma ma ma ma ma ma ma ma Algebra Skills (p. 18) 5. x x 15x 0 18x 18 x x 1 x x 7x 4 7x 189 x 6 x x x 8x 60 7x 490 x 45 x 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) The triangle is obtuse. The triangle is acute 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

15 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. ma ma ma x x 180 x x 164 x 8 The other interior angles each measure x 90 x 59 One of the interior angles is 90 and the other is x 8 1 x 5 0. (x 5) 49 x 44 x 1. 4x 16 x 4. x x x 180 x 10 x x 60 x x 5x 4 4 x 5. x x 10 x x x 500 x 84 x x x 6 16 x 6 56 x 0 x AB ( 0 ) ( 4 0 ) ( ) AB (6 ) ( 4 5 ) 4 ( 9 ) AB ( ( 8 )) ( 7 7 ) AB (0 ( 4 )) ( 6 ( 1 )) AB ( 6 ( )) ( 7 ( 1 )) ( 4 ) ( 6 ) AB ( 8 ) ( 4 ( )) ( 1 0 ) Geometry, Concepts and Skills 65

16 4. AB ( 9 ) ( 6 1 ) ( 1 ) ( 7 ) AB (0 5 ) ( 6 4 ) ( 5 ) right obtuse acute acute 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 KP KM (10) 80; KP PM KM 80 PM 10 PM 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 CD CE 8 CE (8) CE 4 CE; CD DE CE 8 DE 4 DE 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 , , and Yes, the side lengths can form a triangle because 1 5, 1 5, and No, the side lengths can not form a triangle because No, the side lengths can not form a triangle because Yes, the side lengths can form a triangle because , , and Yes, the side lengths can form a triangle because 4, 4, and No, the side lengths can not form a triangle because Yes, the side lengths can form a triangle because and No, the side lengths can not form a triangle because Geometry, Concepts and Skills

17 Chapter 4 Test (p. 4) 1. T JLM. T JKL. T JKL 4. T JLM 5. x (x ) (x 15) 180 6x x 198 x 6. 1x x 15 1x 1 x 5 x x x x 96 x PQ ( 6 0 ) ( 8 0 ) ( 6 ) ( 8 ) (, 6) y P(0, 0) x ( 6, 8) 1 PQ ( ( )) ( 4 6 ) 4 ( ) acute obtuse obtuse y 1 P(, 4) x 14. DC EC () ; DC DE EC DE DE mab maa, so AC BC. maa mac, so BC AB. AC ****, CB ****, BA **** 16. ma ma 46 ; ma 1 ; 1 ma ma ma ma1 180 ma No, the side lengths can not form a triangle because Yes, the side lengths can form a triangle because , , and No, the side lengths can not form a triangle because Yes, the side lengths can form a triangle because , , and Chapter 4 Standardized Test (p. 5) 1. D. H; mabcd A; F 5. C; JK (8 ) ( 5 ) 5 ( 7 ) G; B; x x acute 8. x y , so x y. 10. G Chapter 4 Algebra Review (p. 7) 9 hours hours 4 8 Geometry, Concepts and Skills 67

18 . 10 inches 10 inches feet 1 inches 1 0 inches inches minutes 40 minutes 4 hours 4 60 minutes 40 minutes minutes pounds ounces 0 ounces 0 ounces 9 6 ounces ounces ho 8 da urs ys 5 6 hours 8 days hours 1 day 7 hours/day mil hou es rs 6 0 mile hour s s 0 miles 1 hour 0 miles/hour $ 8 $ 8 4 $ hours 4 hours 4 1 hour $9.50/hour $ 5.88 $ $ bagels 1 bagels 1 1 bagel $.49/bagel 68 Geometry, Concepts and Skills

Chapter 6. Worked-Out Solutions AB 3.61 AC 5.10 BC = 5

Chapter 6. Worked-Out Solutions AB 3.61 AC 5.10 BC = 5 27. onstruct a line ( DF ) with midpoint P parallel to and twice the length of QR. onstruct a line ( EF ) with midpoint R parallel to and twice the length of QP. onstruct a line ( DE ) with midpoint Q