458 CHAPTER 8 Geometry Angles are classified by their degree measurement, as shown in the following summary. A right angle has a measure of 90°, an acute angle has a measure less than 90°, an obtuse angle has a measure greater than 90° but less than 180°, and a straight angle has a measure of 180°. 100 110 120 130 140 150 160 170 80 70 60 40 30 20 10 10 20 30 40 50 60 70 80 100 110 120 140 150 160 170 x 2 1 90 4 5 50 130 Figure 8.8 A B C D E F G Figure 8.9 Example 3 Working with Angles Refer to Fig. 8.9. Determine the following. a) h h BE BC < b) gAC CBG > c) CBE CBG > d) EBF DBG > Solution a) = h h BE BC CBE < or EBC b) = g h AC CBG BC > c) = h CBE CBG BC > d) { } = EBF DBG B > 7 Now try Exercise 37 Did You Know? Classical Geometry Problems Have you ever spent several hours trying to solve a difficult homework problem? Can you imagine studying the same problem for your entire life? Entire generations of mathematicians throughout history spent their entire lives studying three geometry construction problems that originated in ancient Greece. The three problems are (1) trisecting an angle , (2) squaring the circle , and (3) doubling the cube . Trisecting an angle refers to dividing a given angle into three equal angles. Squaring the circle refers to constructing a square that has the exact same area as the area of a given circle. Doubling the cube refers to constructing a cube that has exactly double the volume of a given cube. Eventually, mathematicians came to realize that if one is limited to using only an unmarked straightedge and a compass, then these constructions are impossible. Finally, in 1837 French mathematician Pierre Wantzel proved the impossibility of the first two constructions, and in 1882 German mathematician Carl Lindemann proved the impossibility of the third construction. Right Angle Acute Angle Obtuse Angle Straight Angle x x x x = x 90° < < x 0° 90° < < x 90° 180° = x 180° The symbol is used to indicate right angles. D C A B Figure 8.10 Two angles in the same plane are adjacent angles when they have a common vertex and a common side but no common interior points. In Fig. 8.10, DBC and CBA are adjacent angles, but DBA and CBA are not adjacent angles. Two angles are called complementary angles if the sum of their measures is 90°. Two angles are called supplementary angles if the sum of their measures is 180°.
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