1.5 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Describe Angle Pair Relationships

1.5 Warm-Up 1. The sum of two numbers is 90 and one number is 4 times the other. Write an equation and solve to find the numbers. ANSWER x + 4x = 90; 18, Find m ABD. What kind of angle is it? ANSWER 180°, straight

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

1.5 Guided Practice 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

1.5 Guided Practice 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

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

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

1.5 Example 4 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.

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

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

1.5 Guided Practice 3. Given that 1 is a complement of 2 and m 2 = 8 o, find m o ANSWER 4. Given that 3 is a supplement of 4 and m 3 = 117 o, find m o 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 26 o, 64 o

1.5 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER Example 4 SOLUTION To find vertical angles, look or angles formed by intersecting lines. To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. Identify all of the linear pairs and all of the vertical angles in the figure at the right. 1 and 5 are vertical angles. ANSWER

1.5 Example 5 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. 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

1.5 Example 5 SOLUTION 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 x o + 5x o = 180 o 6x = 180 x = 30 o Write an equation. Combine like terms. Divide each side by 6. The measures of the angles are 30 o and 5(30) o = 150 o. ANSWER

1.5 Guided Practice ANSWER 1 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. Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain. 6.

1.5 Guided Practice 7. The measure of an angle is twice the measure of its complement. Find the measure of each angle. ANSWER 60°, 30°

1.5 Lesson Quiz 1. 1 and 2 are supplementary. If m 1 = 97, find m 2. o 2. 3 and 4 are complementary angles. If m 3= 74, Find m 4. o ANSWER 83 o ANSWER 16 o

1.5 Lesson Quiz ANSWER 96 o ANSWER 36 o 3. Find m EFH. 4. Find m ABC.

1.5 Lesson Quiz 5. Is it possible to draw a figure that contains exactly one pair of vertical angles? Explain. No; once you have drawn a pair of vertical angles, you have drawn two pairs of opposite rays. This automatically gives another pair of vertical angles. ANSWER