8.1 Points, Lines, Planes, and Angles 459 D C A B E 318 Figure 8.11 Example 4 Determining Complementary and Supplementary Angles In Fig. 8.11, we see that = m ABC 31°. a) ABC and CBD are complementary angles. Determine m CBD. b) ABC and CBE are supplementary angles. Determine m CBE. Solution a) The sum of two complementary angles must be 90°, so + = + = = − = m ABC m CBD m CBD m CBD 90° 31° 90° 90° 31° 59° b) The sum of two supplementary angles must be 180°, so m ABC m CBE m CBE m CBE 180° 31° 180° 180° 31° 149° + = + = = − = Now try Exercise 53 Subtract 31° from each side of the equation. Subtract 31° from each side of the equation. 7 Example 5 Determining Complementary Angles If ABC and CBD are complementary angles and m ABC is 26° less than m CBD, determine the measure of each angle (Fig. 8.12). Solution Let = m CBD x. Then = − m ABC x 26, since it is 26° less than m CBD. Because these angles are complementary, we have + = + − = − = = = m CBD m ABC x x x x x 90° ( 26°) 90° 2 26° 90° 2 116° 58° Therefore, = m CBD 58° and = − m ABC 58° 26°, or 32°. Note that + = 58° 32° 90°, which is what we expected. 7 Now try Exercise 67 C D B A Figure 8.12 Example 6 Determining Supplementary Angles If ABC and ABD are supplementary angles and m ABC is five times greater than m ABD, determine m ABC and m ABD (Fig. 8.13). Solution Let = m ABD x, then = m ABC x5 . Since these angles are supplementary, we have + = ° + = ° = ° = ° m ABD m ABC x x x x 180 5 180 6 180 30 Thus, = m ABD 30° and = = m ABC 5(30°) 150°. Note that + = 30° 150° 180°, which is what we expected. 7 Now try Exercise 69 A C D B Figure 8.13
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