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Chapter 1.5

Chapter 1.5. Describe Angle Pair Relationships. Key Terms: Complementary angles Supplementary angles Adjacent angles Linear pair Vertical angles. 2. 1. 3. 5. 4. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

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Chapter 1.5

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  1. Chapter 1.5 Describe Angle Pair Relationships • Key Terms: • Complementary angles • Supplementary angles • Adjacent angles • Linear pair • Vertical angles 2 1 3 5 4

  2. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. Because 32°+ 58° = 90°, BACand RSTare complementary angles. Because 122° + 58° = 180°,CADand RSTare supplementary angles. Because BACand CADshare a common vertex and side, theyare adjacent. EXAMPLE 1 Identify complements and supplements SOLUTION

  3. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1. Because 41° + 49° = 90°, FGK and GKLare complementary angles. Because 49° + 131° = 180°,HGKand GKL are supplementary angles. Because FGKand HGKshare a common vertex and side, theyare adjacent. for Example 1 GUIDED PRACTICE

  4. Are KGHand LKGadjacent angles ? Are FGKand FGHadjacent angles? Explain. 2. KGH and LKG do not share a common vertex , they are not adjacent. FGK and FGH have common interior points, they are not adjacent. for Example 1 GUIDED PRACTICE

  5. 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 • Given that 1 is a complement of 2 and m1 = 68°, • find m2. SOLUTION

  6. b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124° b. Given that 3 is a supplement of 4and m 4=56°, find m3. EXAMPLE 2 Find measures of a complement and a supplement SOLUTION

  7. Sports When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find mBCEand mECD. EXAMPLE 3 Find angle measures

  8. Use the fact that the sum of the measures of supplementary angles is 180°. STEP1 mBCE+m∠ ECD=180° EXAMPLE 3 Find angle measures SOLUTION Write equation. (4x+ 8)°+ (x +2)°= 180° Substitute. 5x + 10 = 180 Combine like terms. 5x = 170 Subtract10 from each side. x = 34 Divide each side by 5.

  9. STEP2 Evaluate: the original expressions when x = 34. m BCE = (4x + 8)° = (4 34 + 8)° = 144° m ECD = (x + 2)° = ( 34 + 2)° = 36° ANSWER The angle measures are144°and36°. EXAMPLE 3 Find angle measures

  10. 3. Given that 1 is a complement of 2 and m2 = 8° , find m1. m 1 = 90° – m 2 = 90°– 8° = 82° 1 8° 2 for Examples 2 and 3 GUIDED PRACTICE SOLUTION You can draw a diagram with complementary adjacent angle to illustrate the relationship

  11. 4. Given that 3 is a supplement of 4 and m3 = 117°, find m4. m 4 = 180° – m 3 = 180°– 117° = 63° 117° 3 4 for Examples 2 and 3 GUIDED PRACTICE SOLUTION You can draw a diagram with supplementary adjacent angle to illustrate the relationship

  12. m LMN + m PQR = 90° for Examples 2 and 3 GUIDED PRACTICE 5.LMNand PQRare complementary angles. Find the measures of the angles if m LMN= (4x –2)° and m PQR = (9x + 1)°. SOLUTION Complementary angle (4x – 2 )° + ( 9x + 1 )° = 90° Substitute value 13x – 1 = 90 Combine like terms 13x = 91 Add 1 to each side x = 7 Divide 13 from each side

  13. m LMN = (4x – 2 )° = (4·7 – 2 )° = 26° m PQR = (9x – 1 )° = (9·7 + 1)° = 64° m PQR ANSWER m LMN = 64° = 26° for Examples 2 and 3 GUIDED PRACTICE Evaluate the original expression whenx = 7

  14. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. Identify all of the linear pairs and all of the vertical angles in the figure at the right. ANSWER 1 and 5 are vertical angles. ANSWER EXAMPLE 4 Identify angle pairs 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.

  15. ALGEBRA 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. 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. EXAMPLE 5 Find angle measures in a linear pair SOLUTION

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

  17. 6. Do any of the numbered angles in the diagram below form a linear pair?Which angles are vertical angles? Explain. No, adjacent angles have their non common 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 For Examples 4 and 5 GUIDED PRACTICE ANSWER

  18. The measure of the angles are 30° and 2( 30 )° = 60° ANSWER For Examples 4 and 5 GUIDED PRACTICE 7. The measure of an angle is twice the measure of its complement. Find the measure of each angle. SOLUTION Let x° be the measure of one angle . The measure of the other angle is 2x° then use the fact that the angles and their complement are complementary to write an equation x° + 2x° = 90° Write an equation 3x = 90 Combine like terms x = 30 Divide each side by 3

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