EXAMPLE 1 Identify complements and supplements In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. SOLUTION Because 32°+ 58° = 90°, BAC and RST are complementary angles. Because 122° + 58° = 180°, CAD and RST are supplementary angles. Because BAC and CAD share a common vertex and side, they are adjacent.

GUIDED PRACTICE for Example 1 In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1. FGK and GKL, HGK and GKL, FGK and HGK ANSWER

GUIDED PRACTICE for Example 1 Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain. 2. No, they do not share a common vertex. No, they have common interior points. ANSWER

EXAMPLE 2 Find measures of a complement and a supplement Given that 1 is a complement of 2 and m 1 = 68°, find m 2. SOLUTION a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 2 = 90° – m 1 = 90° – 68° = 22°

EXAMPLE 2 Find measures of a complement and a supplement b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3. SOLUTION b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°

EXAMPLE 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. SOLUTION To find vertical angles, look or angles formed by intersecting lines. 1 and 5 are vertical angles. ANSWER To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER

EXAMPLE 5 Find angle measures in a linear pair Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle. ALGEBRA SOLUTION Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.

Find angle measures in a linear pair EXAMPLE 5 Find angle measures in a linear pair xo + 5xo = 180o Write an equation. 6x = 180 Combine like terms. x = 30o Divide each side by 6. The measures of the angles are 30o and 5(30)o = 150o. ANSWER

GUIDED PRACTICE For Examples 4 and 5 Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain. 6. ANSWER No, no adjacent angles have their noncommon sides as opposite rays, 1 and 4 , 2 and 5, 3 and 6, these pairs of angles have sides that from two pairs of opposite rays.

GUIDED PRACTICE For Examples 4 and 5 7. The measure of an angle is twice the measure of its complement. Find the measure of each angle. ANSWER 60°, 30°

EXAMPLE 3 Find angle measures Sports When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

EXAMPLE 3 Find angle measures SOLUTION STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°. m BCE + m ∠ ECD = 180° Write equation. (4x + 8)° + (x + 2)° = 180° Substitute. 5x + 10 = 180 Combine like terms. 5x = 170 Subtract 10 from each side. x = 34 Divide each side by 5.

EXAMPLE 3 Find angle measures SOLUTION STEP 2 Evaluate: the original expressions when x = 34. m BCE = (4x + 8)° = (4 34 + 8)° = 144° m ECD = (x + 2)° = ( 34 + 2)° = 36° The angle measures are 144° and 36°. ANSWER

GUIDED PRACTICE for Examples 2 and 3 3. Given that 1 is a complement of 2 and m 2 = 8o, find m 1. 82o ANSWER 4. Given that 3 is a supplement of 4 and m 3 = 117o, find m 4. 63o ANSWER 5. LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)o and m PQR = (9x + 1)o. ANSWER 26o, 64o