9 9. M, 0. M ( 9, 4) 7. If WZ XZ, then ZWX ZXW ; Base Angles Theorem (Thm..6). M 9,. M ( 4, ) 74. If XZ XY, then XZY Y; Base Angles Theorem (Thm..6). M, 4. M ( 9, ) 7. If V WZV, then WV WZ; Converse of Base Angles Theorem (Thm..7). 7 M, 6. M, 76. If ZV ZY, then V Y; Base Angles Theorem (Thm..6) 7. M, 8. 7 M, 77. If ZWX ZXW, then ZW ZX; Converse of Base Angles Theorem (Thm..7) 9. M (, ) 40. 4. M 7, M 7, 6 4. M (, ) 4. 8 44. 8 4. 6 46. 0 47. 97 48. 49 49. 97 0. 6. 09. 78. If XZY Y, then ZX YX; Converse of Base Angles Theorem (Thm..7) 79. If V Y, then ZV ZY; Converse of Base Angles Theorem (Thm..7) 80. 6 8. 9 8. 0 8. 84. 8. 86. EF, FG, EG 87. ST, RS, RT Chapter 9 9. Start Thinking A = a + b. 4. 7. 9 6. 8 7. 8. XV 9. YW 60. YZ 6. XZ 6. ZW and ZY, or ZV and ZX 64. Sample answer: ZV and ZW 6. ZV, ZW, ZX, ZY 6. point M; 6 66. line s; 6 67. Mk ; 68. Mm ;0 69. line s; 84 70. point M; 4 7. Mm;90 7. Mk;0 a A = c ; Because the area of the original diagram must equal the area of the reassembled diagram, a + b = c, which is a statement of the Pythagorean Theorem (Thm. 9.). You have proved the theorem with your construction. 9. Warm Up. x = ±. x = ± 4. b c c a b x =± 4. x = ± 9 6 A9
. x = ± 9 6. x = ± 6 9. Cumulative Review Warm Up. P = + 6 units. P = 0 units. P = + + units 4. P = 4 units 9. Practice A. ; yes. 7 ;no. ; yes 4. ; no. ;no 6. 4; yes 7. no 8. yes 9. yes; right 0. yes; acute. yes; obtuse. yes; obtuse. 60 ft 9. Practice B. 8 ;no. 4; yes. 0; no 4. no. yes 6. yes; 6 + = 9 4. a. no; Sample answer: Let a =, b = 4, and c =. So, a + = 4, b + =, and c + = 6. So,, 4, and form a Pythagorean triple, but 4,, and 6 do not because 4 + 6. b. yes; Sample answer: If a, b, and c form a Pythagorean triple, a + b = c is true. Multiply each side by 4 to get the equation 4a + 4b = 4 c. This is equivalent to ( a) ( b) ( c) + =. So by definition, a, b, and c also form a Pythagorean triple. c. no; Sample answer: Let a =, b = 4, and c =. So a = 9, b = 6, and c =. So,, 4, and form a Pythagorean triple, but 9, 6, and do not because 9 + 6. d. no; Sample answer: Let a =, b = 4, and c =. So, a =, b =, and c =. So,, 4, and form a Pythagorean triple, but,, and do not because ( ) + ( ).. 6. 4 9 in. 7. 44 beads 9. Puzzle Time A SCREENSAVER 9. Start Thinking 7. yes; right 8. yes; obtuse 9. yes; acute 0. yes; 6ft < x < 0ft s s s. a..94 ft b. 88 6.97 ft 9. Enrichment and Extension. EF = 6, EC = 4, FC = EF = x, EC = 0 x, FC = x 0 x,. ( ) area of EFC = ( 0 x) x ( 0 x) s s s, s, and ; s no; The side lengths cannot all be integer values that satisfy the equation a + b = c. 9. Warm Up s. x = 4.. x = 4 7 4. x = x = 6. k = 7, k =, k = 8, and k =. x = 6. x = 4. = A9
9. Cumulative Review Warm Up. 8 9. Practice A. 7. 4.. 4.. 4. x =, y =. x = 4, y = 8 6. x =, y = 7. 8. 60. m 7. yd 9. 6 ft; 8. ft; 0.4 ft 9. Practice B... a. 0-60 -90 triangle b. neither c. 4-4 -90 triangle 4. x = 4, y =. x =, y = 4 6. x = 8, y = 8, z = 8 7. about x 4 y 4 4 a 8 b 9 8 c 6 6 0 0.6 ft 9. Enrichment and Extension. s =, v =, w =, x =, y = 0, z = 0 +. about 4.094 units. 7 4. about 8.9 units. 4 points; (,, ) (,, ) (,, ) (, ) 6. VW.4, VX 9.80, WX 9.46 7. BC = 0., CD.7, BD.80 9. Puzzle Time A TURTLE 9. Start Thinking. Both ABC and ACD contain A, which is congruent to itself by the Reflexive Property of Angle Congruence (Thm..). Both triangles contain a right angle, and all right angles are congruent. So, the two triangles are similar by the AA Similarity Theorem (Thm. 8.).. Both ABC and CBD contain B which is congruent to itself by the Reflexive Property of Angle Congruence (Thm..). Both triangles contain a right angle, and all right angles are congruent. So, the two triangles are similar by the AA Similarity Theorem (Thm. 8.).. By the transitive property, you can conclude that ACD CBD. 9. Warm Up. x = 0. x =. 7 9. Cumulative Review Warm Up. m =, m = 48. m = 9, m = 8. m = 67, m = 9. Practice A. KLM JLK JKM. YXU ZXY ZYU. 4.8 4. about 4.. about 7. 6. 6 7. 4 8. 4 9. 40 0. x = 4. x = 6.. about 67.8 cm 9. Practice B. CBD ABC ACD. CB. CD = 8, AD = 0, AC = A94
4. 4. 0 6. x = 7. 8. 0 9. 0. w =. x = 6, y =, z = 9.4 Warm Up. x. y y = 6, = x y = 8, = x 4 8 6 6. 7 x =, y =, z = 4 4 4. a. 48 in. b..6 in. c. The support attaches about inches from the top of the plywood; It divides the plywood into pieces measuring approximately inches and 4 inches.. x y 9 =, = x 9.4 Cumulative Review Warm Up. 9. Enrichment and Extension. 6. DC = 7, BD =, AB =. a. b. 8.4 c. yes; When you compute the harmonic mean using 4x and x, you get an answer of 6x. = =. D ( 0, 0) 4. PR.; QS 6. 6 D, 7. 9. Puzzle Time BECAUSE HE WANTED TO TURN OVER A NEW LEAF. 9.4 Start Thinking Sample answer: x in., y in., x in., 8 y in., x in., y 4 in. 8 y. x =.7. y x =.7 y. x =.7 It appears that regardless of the size of the 0-60 -90 triangle, the ratios of corresponding sides are equal or approximately equal. A9
.. Sample answer: 4 4 4... 8.6 6. x = 8, y 4. 7. 64 ft 8..0 units 9.4 Enrichment and Extension. AB = BC = CD = DA = 89, m ABC = ADC 64, m DAB = BCD 6 ; The diagonals of a rhombus bisect each other and intersect at a right angle to form four congruent right triangles. The Pythagorean Theorem (Thm. 9.) is used to find the side lengths, and the tangent ratio is used to compute the measures of the angles of the rhombus. a = = b. tan x ; tan( 90 x ) b a 9.4 Practice A. tan S = 0.467, tan R =.4. tangent ratio should be the ratio of the opposite side to the adjacent side, not the adjacent to the opposite; tan k = 4.. 4. 8.. 7.9 6. 64.0 7. 9 8. 9. m 9.4 Practice B.. 7 4 tan J = 0.97, tan K =.486 4 7 tan D =.884, tan E = 0.6 4. They are reciprocals of each other. 4. The relationship will be true in all cases except when x = 90 or x = 0; This is because tan 0 = 0 and tan 90 is undefined...96 in. 6. 8.7 7. tan 0 = and tan 60 = ; If a = b = 0, tan a + tan b = + =, but ( a b ) ( ) So, a + b ( a + b ) tan + = tan 0 + 0 = tan 60 =. tan tan tan. 8. 8.9 9.4 Puzzle Time A PEST TEST A96
9. Start Thinking Sample answer: Because the hypotenuse of a right triangle is always the longest side, the sine and cosine ratios of an acute angle will always be less than one. Because the length of each leg is always greater than zero, the sine and cosine ratios of an acute angle will always be greater than zero. As the acute angle gets larger, the sine ratio will get closer to one, and the cosine ratio will get closer to zero. 9. Warm Up. about 4.4. about 7.8. about.9 9. Cumulative Review Warm Up. ( 6,. ).. (., 7 ) 4. 9. Practice A.. 4, 6 7, 7 7 sin J = 0.9, sin K = 0.846, cos J = 0.846, cos K = 0.9 sin J 8 = 0.884, sin K = 7 7 0.4706, cos J 8 = 0.4706, cos K = 7 7 0.884. cos 68 4. sin 4. sin 7 6. cos 7. a 4.7, b 0. 8. x 8., y 7.0 9. r 4., s 4.6 0. B; Because sin A is the ratio of the length of the leg opposite A to the length of the hypotenuse, and the hypotenuse is the longest side of a right triangle, the value of sin A must be less than.. 69 ft 9. Practice B. sin R = 4 7 = 0.96, sin S = = 0.8, cos R = 7 4 = 0.8, cos S = = 0.96. 9 7 sin R = 0.897, sin S = 0.440, 6 6 7 9 cos R = 0.440, cos S = 0.897 6 6. cos 8 4. sin 9. sin 60 cos 60 6. p., q 4. 7. x.6, y 6.6 8. a 9., b 6.7 9. 4 cm 0. a. 6. ft b. ft c. ft 9. Enrichment and Extension. x =.999. w =.94 cm, x =.4 cm, y = 6.66 cm, z = 9.49 cm. 70 units 4. 9. in.. : 6. ( sin a ) ( cos a ) 7. 0.8 9. Puzzle Time x y + = + z z A TERMINAL ILLNESS 9.6 Start Thinking x + y z = = = z z. 60. 4. 0 4. 0. 60 6. 4 9.6 Warm Up... 7 7 ; sin θ =, cos θ =, tan θ = 0 0 709 709 709; sin θ =, cos θ =, 709 709 tan θ = 9 0 9 ; sin θ =, cos θ =, 9 9 0 9 tan θ = 87 A97
9.6 Cumulative Review Warm Up. yes; You can prove ABC XYZ by the AAS Congruence Theorem (Thm..).. no; You cannot prove ABC XYZ. However, you could prove that they are similar by the AA Similarity Theorem (Thm 8.).. yes; You can prove ABC XYZ. Because of the perpendicular segments, you know both triangles are right, so by the HL Congruence Theorem (Thm..9), you can prove they are congruent. 9.6 Practice A. R. R. Q 4. 9.8. 8. 6. 6.4 7. m P =, QR 7.6, PR. 8. m D 4.7, DF 0.6, m E 47. 9. m A = 9, BC., AB 8.0 0. 499 ft. 7.4 9.6 Practice B. W. X. 4. 4. 87.. 79. 6. m T = 66, ST 6., RT 4.7 7. m E 4.4, EF = 9, m D 48.6 8. m P 7.7, PQ.9, m R 7. 9. a. about 09 ft b. about 648 ft c. about 609 ft 0. a. about 0. mi b. about 0.8 mi c. about 7.. 0 6. 44.4 7. 4.9 8. 0.9 9.6 Puzzle Time MISTAKES 9.7 Start Thinking... All three ratios are the same; Sample answer: yes 9.7 Warm Up. a 0.9. c.. b 0. 4. B 7.8. A 67.8 6. C 49. 9.7 Cumulative Review Warm Up. m X =, m Z =. m X =, m Z =. m X =, m Z = 4 9.7 Practice A. 0.46..40. 0.99 4. 6..8 m. 4.9 ft Law of Sines Law of Cosines Neither AAS, ASA, SSA SSS, SAS 7. m B = 0, a.9, c 6.7 8. m A 40.8, m B 60.6, m C 78.6 9. m A 8.6, b 7., m C 0.4 0. a 90., m A 6., m B 6.9. m A 48., m B 06.6, m C.. m B = 74, b 4., c 4.0 AAA 9.6 Enrichment and Extension. a. 8 b. 40 c.. 7. 9.7 Practice B..074. 0.67. 0.9998. csc θ =. sec θ = 4. cot θ = 4. 60.4 cm..7 in. A98
6. Draw a diagonal to form the SAS case of a triangle, use the area formula A = ab sin C, and then double this area; A 678 m 7. m B =, b 8., c. 8. a.0, m B 6.4, m C 86.6 9. m B 64., c.4, m C 7. 0. m A 6., m B 4., m C 00.. a.6, b., m C = 7. m A 9., m B.4, m C 8.0. a. about 9.7 in. b. about 8.8 9.7 Enrichment and Extension. A 69.7, B 0., c 4.6. C = 78.9, B = 9., a =.6. a 8.9, B 8.9, C 7. 9.7 Puzzle Time BECAUSE THEN YOU LL BE A MILE AWAY AND YOU LL HAVE THEIR SHOES Cumulative Review. 6.. 9 4. 0. 9 6. 8 7 7. 8. 4 7 9. 0. 9.. 4 7 7 4.. 7. 8 9. x = 80 0. x =. x =. x =. x = 04 4. x = 87. x = 9 6. x = 7. x = 6 8. x = 9. x =. 0. x = 7.. x =. x =. x = 4. x = 8. x = 66 6. x = 9 7. x = 6 8. x = 6 9. x =. 40. x = 6 4. x = 4 4. x = 4. x = 44. x = 7 4. x = 46. r = ft, d = 0 ft 47. r = ft, d = 6 ft 48. r = 8 in., d = 6 in. 49. r = 7 ft, d = 4 ft 0. r = 4 in., d = 8 in.. r = in., d = in.. 4. 94 4. 68. 6. 7. 8 8. acute isosceles triangle 9. right isosceles triangle 60. equilateral triangle 6. right scalene triangle 6. x =, y = 8 6. r = 9, s = 0 64. m = 4, x = 6. a = 68, k = 8 66. a. b. 67. 0 68. 40 69. 0 70. 4 7. 7. 6 7. 4 74. 7 6. 6 7. 8. 9 7. a. 6 in. b. + 6 in. c. 0. in. A99