8.1 Points, Lines, Planes, and Angles 463 In Exercises 71–74, parallel lines are cut by the transversal shown. Determine the measures of 1 through 7. 71. 1 2 3 4 5 6 7 1308 * 72. 1 2 3 4 5 6 120 7 8 * 73. 1 2 3 4 5 6 7 208 * 74. 1 2 3 4 5 6 7 258 * In Exercises 75–78, the angles are complementary angles. Determine the measures of 1 and 2. 75. 1 2 x 2x – 9 = = m m 1 33°, 2 57° 76. 1 2 x 5x – 6 = = m m 1 16°, 2 74° 77. 2x + 12 x 2 1 = = m m 1 64°, 2 26° 78. 8x – 9 1 2 x = = m m 1 79°, 2 11° In Exercises 79–82, the angles are supplementary angles. Determine the measures of 1 and 2. 79. 2 1 x 3x – 4 = = m m 1 134°, 2 46° 80. 1 2 x 7x – 12 = = m m 1 156°, 2 24° 81. 2 1 x 5x + 6 = = m m 1 29°, 2 151° 82. 2 1 x 6x + 5 = = m m 1 25°, 2 155° In Exercises 83–90, the following figure suggests several lines and planes. The lines may be described by naming two points, and the planes may be described by naming three points. Use the figure to name the following. Many answers are possible. A B C D E F G H I J 83. Two parallel lines g g EF DG and 84. Two skew lines g g EF GH and 85. Two parallel planes Plane ABG and plane JCD 86. Two planes that intersect at right angles Plane ABG and plane BCD 87. Three planes whose intersection is a single point { } = > > ABG ABC BCD B Plane plane plane 88. Three planes whose intersection is a line = g HGD FGD BGD GD Plane plane plane > > 89. A line and a plane whose intersection is a point { } = gBC ABG B plane > 90. A line and a plane whose intersection is a line = g g AB ABG AB plane > Concept/Writing Exercises In Exercises 91–96, determine whether the statement is always true, sometimes true, or never true. Explain your answer. 91. Two lines that are both parallel to a third line must be parallel to each other. Always true. If any two lines are parallel to a third line, then they must be parallel to each other. 92. A triangle contains exactly two acute angles. * 93. Vertical angles are complementary angles. Sometimes true. Vertical angles are only complementary when each is equal to 45°. 94. Alternate exterior angles are supplementary angles. * 95. Alternate interior angles are complementary angles. * 96. A triangle contains two obtuse angles. Never true. The sum of two obtuse angles is greater than 180°. 97. a) How many lines can be drawn through a given point? An infinite number b) How many planes can be drawn through a given point? An infinite number 98. What is the intersection of two distinct nonparallel planes? A line 99. How many planes can be drawn through a given line? An infinite number 100. a) Will three noncollinear points A, B, and C always determine a plane? Explain. Yes b) Is it possible to determine more than one plane with three noncollinear points? Explain. No c) How many planes can be constructed through three collinear points? An infinite number Challenge Problems/Group Activities 101. Geometric Construction Use a straightedge and a compass to construct a triangle with sides of equal length (an equilateral triangle) by doing the following: a) Use the straightedge to draw a line segment of any length and label the end points A and B (Fig. a). *See Instructor Answer Appendix
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