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Understanding Complementary and Supplementary Angles: Relationships and Calculations

This guide explores the concepts of complementary and supplementary angles, demonstrating how to find their measures through various examples. It includes detailed descriptions of adjacent angles and their relationships, illustrated by diagrams. Learn how to set up equations based on the definitions of complementary (sum of 90°) and supplementary angles (sum of 180°) with practical examples, including sports applications. The step-by-step solutions will enhance your understanding of angle relationships and problem-solving techniques.

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Understanding Complementary and Supplementary Angles: Relationships and Calculations

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  1. 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

  2. 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

  3. 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

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

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