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Chapter Nine Geometry

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Chapter Nine Geometry CHAPTER NINE GEOMETRY Exercise Set 9.1 1. a) Undefined terms, definitions, postulates (axioms), and theorems b) First, Euclid introduced undefined terms. Second, he introduced certain definitions. Third, he stated primitive propositions called postulates (axioms) about the undefined terms and definitions. Fourth, he proved, using deductive reasoning, other propositions called theorems. 2. An axiom (postulate) is a statement that is accepted as being true on the basis of its "obviousness" and its relation to the physical world. A theorem is a statement that has been proven using undefined terms, definitions, and axioms. 3. Two lines in the same plane that do not intersect are parallel lines. 4. Two lines that do not lie in the same plane and do not intersect are called skewed lines. 5. Two angles in the same plane are adjacent angles when they have a common vertex and a common side but no common interior points. 6. Two angles the sum of whose measure is 180° are called supplementary angles. 7. Two angles the sum of whose measure is 90° are called complementary angles. 8. An angle whose measure is 180° is a straight angle. 9. An angle whose measure is greater than 90° but less than 180° is an obtuse angle. 10. An angle whose measure is less than 90° is an acute angle. 11. An angle whose measure is 90° is a right angle. 12. In the pair of intersecting lines below, ( 1 and ( 3 are vertical angles as are ( 2 and ( 4. 2 1 3 4 JJJG 13. Half line, 14. Half open line 15. Line segment, AB 16. Ray, AB segment, HJJG JJJG 17. Line, AB 18. Half line, 19. Open line 20. Ray, BA segment, JJJG HJJG HJJG 21. BD 22. EG 23. 24. AD 25. {B, F} 26. {C} 27. {C} 28. 295 296 CHAPTER 9 Geometry HJJG 29. BC 30. +BCF 31. BC 32. ∅ ∅ JJJG HJJG 33. 34. 35. BC 36. DE 37. (ABE 38. (FBE 39. (EBC 40. {B} 41. 42. ∅ 43. 44. {B} 45. Obtuse 46. Straight 47. Straight 48. Acute 49. Right 50. None of these 51. None of these 52. Right 53. 901971°− °= ° 54. 90891°− °= ° °−3 °=1 ° °−1 °=2 ° 55. 9032574 4 56. 9043463 3 57. 90°− 64.7 °= 25.3 ° 58. 90°− 0.01 °= 89.99 ° 59. 180°− 91 °= 89 ° 60. 180°− 8 °= 172 ° 61. 180°− 20.5 °= 159.5 ° 62. 180°− 179.99 °= 0.01 ° °−5 °=2 ° °−79 °= ° 63. 180 437 136 7 64. 180 641616 115 65. d 66. b 67. c 68. f 69. e 70. a 71. Let x = measure of ( 2 72. Let x = measure of ( 1 x+4 = measure of ( 1 90 − x = measure of ( 2 xx++=490 xx−−=()90 62 += 2490x xx−+=90 62 = 286x 29062x −= 86 = xm==°43,( 2 2152x 2 152 xm+=4 43 += 4 47 ° ,( 1 xm==°76 ,( 1 2 90907614,2−=xm − = ° ( 73. Let x = measure of ( 1 74. Let x = measure of ( 1 180 − x = measure of ( 2 17x = measure of ( 2 xx−−=()180 88 xx+=17 180 xx−+=180 88 18x = 180 −= 180 2x 180 88 xm==°10 ,( 1 2268x = 18 ==°() 268 171710170,2xm( xm==°134 ,( 1 2 180−=xm 180 − 134 = 46 ° ,( 2 SECTION 9.1 297 75. m( 1 + 125° = 180° 76. m( 3 + 30° = 180° m( 1 = 55° m( 3 = 150° m( 2 = m( 1 (vertical angles) m( 1 = 30° (vertical angles) m( 3 = 125° (vertical angles) m( 2 = m( 3 (vertical angles) m( 5 = m( 2 (alternate interior angles) m( 4 = m( 1 (corresponding angles) m( 4 = m( 3 (alternate interior angles) m( 7 = m( 4 (vertical angles) m( 7 = m( 4 (vertical angles) m( 6 = m( 3 (alternate interior angles) m( 6 = m( 5 (vertical angles) m( 5 = m( 6 (vertical angles) Measures of angles 3, 4, and 7 are each 125°. Measures of angles 1, 4, and 7 are each 30°. Measures of angles 1, 2, 5, and 6 are each 55°. Measures of angles 2, 3, 5, and 6 are each 150°. 77. m( 1 + 25° = 180° 78. m( 3 + 120° = 180° m( 1 = 155° m( 3 = 60° m( 3 = m( 1 (vertical angles) m( 4 = 120° (vertical angles) m( 2 = 25° (vertical angles) m( 7 = m( 3 (vertical angles) m( 4 = m( 3 (alternate interior angles) m( 6 = m( 3 (alternate interior angles) m( 7 = m( 4 (vertical angles) m( 1 = m( 6 (vertical angles) m( 5 = m( 2 (corresponding angles) m( 5 = m( 4 (alternate exterior angles) m( 6 = m( 5 (vertical angles) m( 2 = m( 5 (vertical angles) Measures of angles 2, 5, and 6 are each 25°. Measures of angles 2, 4, and 5 are each 120°. Measures of angles 1, 3, 4, and 7 are each 155°. Measures of angles 1, 3, 6, and 7 are each 60°. 79. x + 3x + 10 = 90 80. x + 7x + 2 = 90 4x + 10 = 90 8x + 2 = 90 4x = 80 8x = 88 80 88 x = = 20°, m( 2 x = = 11°, m( 1 4 8 3x + 10 = 3(20) + 10 = 70°, m( 1 7x + 2 = 7(11) + 2 = 79°, m( 2 81. x + 2x - 9 = 90 82. x + 8x - 9 = 90 3x - 9 = 90 9x - 9 = 90 3x = 99 9x = 99 99 99 x = = 33°, m( 1 x = = 11°, m( 2 3 9 2x - 9 = 2(33) - 9 = 57°, m( 2 8x - 9 = 8(11) - 9 = 79°, m( 1 83. x + 2x - 15 = 180 84. x + 4x + 10 = 180 3x - 15 = 180 5x + 10 = 180 3x = 195 5x = 170 195 170 x = = 65°, m( 2 x = = 34°, m( 2 3 5 2x - 15 = 2(65) - 15 = 115°, m( 1 4x + 10 = 4(34) + 10 = 146°, m( 1 298 CHAPTER 9 Geometry 85. x + 5x + 6 = 180 86. x + 6x + 5 = 180 6x + 6 = 180 7x + 5 = 180 6x = 174 7x = 175 174 175 x = = 29°, m( 1 x = = 25°, m( 1 6 7 5x + 6 = 5(29) + 6 = 151°, m( 2 6x + 5 = 6(25) + 5 = 155°, m( 2 87. a) An infinite number of lines can be drawn through a given point. b) An infinite number of planes can be drawn through a given point. 88. If the two planes are not parallel, the intersection is a straight line. 89. An infinite number of planes can be drawn through a given line. 90. a) Yes, any three noncollinear points always determine a plane. b) No, the plane determined is unique. c) An infinite number of planes can be drawn through three collinear points. For Exercises 91 - 98, the answers given are one of many possible answers. HJJG HJJJG 91. Plane ABG and plane JCD 92. EF and DG HJJG HJJJG 93. BG and DG 94. Plane ABG and plane BCD HJJG 95. Plane AGB∩∩plane ABCplane BCD ={ B} 96. Plane HGD∩∩=plane FGDplane BGD GD HJJG HJJG HJJG 97. BC∩=plane ABG{ B} 98. AB∩=plane ABG AB 99. Always true. If any two lines are parallel to a third line, then they must be parallel to each other. 100. Sometimes true. A triangle must always contain at least two acute angles. Some triangles contain three acute angles. 101. Sometimes true. Vertical angles are only complementary when each is equal to 45°. 102. Sometimes true. Alternate exterior angles are only supplementary when each is equal to 90°. 103. Sometimes true. Alternate interior angles are only complementary when each is equal to 45°. 104. Never true. The sum of two obtuse angles is greater than 180°. 105. No. Line m and line n may intersect. 106. No. Line l and line n may be parallel or skewed. 107. SECTION 9.2 299 108. 1 4 2 3 mm((1+=°2180 mm((3+=°4 180 180°+ 180 °= 360 ° 109. a) D C E B A Other answers are possible. b) Let m((ABC== x and m CBD y. xy+=90 ° and y =2 x Substitute yx=+=°2 into xy90 . xx+=°290 390x =° ° 390x = 33 x=°=30 m( ABC c) m(CBD= y yx==2 230() °=° 60 d) m(( ABD+=° m DBE 180 m( ABD=+= x y 306090. °+°=° 90°+mDBE( = 180 ° mDBE( =°−°=°180 90 90 . Exercise Set 9.2 1. A polygon is a closed figure in a plane determined by three or more straight line segments. 2. A regular polygon is one whose sides are all the same length and whose interior angles all have the same measure; other polygons may have sides of different length and interior angles with different meaures. 3. The different types of triangles are acute, obtuse, right, isosceles, equilateral, and scalene. Descriptions will vary. 4. The different types of quadrilaterals are trapezoid, parallelogram, rhombus, rectangle, and square. Descriptions will vary. 5. If the corresponding sides of two similar figures are the same length, the figures are congruent figures. 6. Figures that have the same shape but may be of different sizes are similar figures. 7. a) Rectangle 8. a) Triangle 9. a) Hexagon 10. a) Octagon b) Not regular b) Regular b) Regular b) Not regular 11. a) Rhombus 12. a) Pentagon 13. a) Octagon 14. a) Dodecagon b) Not regular b) Regular b) Not regular b) Not regular 300 CHAPTER 9 Geometry 15. a) Scalene 16. a) Isosceles 17. a) Isosceles 18. a) Isosceles b) Right b) Acute b) Obtuse b) Right 19. a) Equilateral 20. a) Scalene 21. a) Scalene 22. a) Scalene b) Acute b) Acute b) Obtuse b) Right 23. Parallelogram 24. Rectangle 25. Rhombus 26. Trapezoid 27. Trapezoid 28. Square 29. The measures of the other two angles of the triangle are 138° and 25° (by vertical angles). Therefore, the measure of angle x is 180° - 138° - 25° = 17°.
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