Special Pairs of Angles 2-4

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. Because 41° + 49° = 90°, FGK and GKL are complementary angles. Because 49° + 131° = 180°, HGK and GKL are supplementary angles. Because FGK and HGK share a common vertex and side, they are adjacent.

GUIDED PRACTICE for Example 1 Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent 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.

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 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 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

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

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

GUIDED PRACTICE for Examples 2 and 3 5. LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)° and m PQR = (9x + 1)°. SOLUTION m LMN + m PQR = 90° 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

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

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

The measure of the angles are 30° and 2( 30 )° = 60° 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. 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 ANSWER The measure of the angles are 30° and 2( 30 )° = 60°

EXAMPLE 2 Name the property shown Name the property illustrated by the statement. a. If R T and T P, then R P. b. If NK BD , then BD NK . SOLUTION Transitive Property of Angle Congruence a. b. Symmetric Property of Segment Congruence

GUIDED PRACTICE for Example 2 2. CD CD Reflexive Property of Congruence ANSWER 3. If Q V, then V Q. Symmetric Property of Congruence ANSWER

EXAMPLE 3 Prove the Vertical Angles Congruence Theorem Prove vertical angles are congruent. GIVEN: 5 and 7 are vertical angles. PROVE: ∠ 5 ∠ 7

Prove the Vertical Angles Congruence Theorem EXAMPLE 3 Prove the Vertical Angles Congruence Theorem STATEMENT REASONS 5 and 7 are vertical angles. 1. 1. Given 2. 5 and 6 are a linear pair. 6 and 7 are a linear pair. 2. Definition of linear pair, as shown in the diagram 3. 5 and 6 are supplementary. 6 and 7 are supplementary. 3. Linear Pair Postulate 4. ∠ 5 ∠ 7 Congruent Supplements Theorem 4.

GUIDED PRACTICE for Example 3 In Exercises 3–5, use the diagram. 3. If m 1 = 112°, find m 2, m 3, and m 4. ANSWER m 2 = 68° m 3 = 112° m 4 = 68°

GUIDED PRACTICE for Example 3 4. If m 2 = 67°, find m 1, m 3, and m 4. ANSWER m 1 = 113° m 3 = 113° m 4 = 67° 5. If m 4 = 71°, find m 1, m 2, and m 3. ANSWER m 1 = 109° m 2 = 71° m 3 = 109°

GUIDED PRACTICE for Example 3 6. Which previously proven theorem is used in Example 3 as a reason? Congruent Supplements Theorem ANSWER

Use right angle congruence EXAMPLE 1 Use right angle congruence Write a proof. GIVEN: AB BC , DC BC PROVE: B C STATEMENT REASONS 1. AB BC , DC BC 1. Given 2. B and C are right angles. 2. Definition of perpendicular lines 3. B C 3. Right Angles Congruence Theorem

EXAMPLE 2 Prove a case of Congruent Supplements Theorem Prove that two angles supplementary to the same angle are congruent. GIVEN: 1 and 2 are supplements. 3 and 2 are supplements. PROVE: 3

Prove a case of Congruent Supplements Theorem EXAMPLE 2 Prove a case of Congruent Supplements Theorem STATEMENT REASONS 1. 3 and 2 are supplements. 1 and 2 are supplements. Given 1. 2. m 1+ m 2 = 180° m 3+ m 2 = 180° 2. Definition of supplementary angles 3. m 1 + m 2 = m 3 + m 2 Transitive Property of Equality 3. 4. m 1 = m 3 Subtraction Property of Equality 4. 5. 3 Definition of congruent angles 5.

GUIDED PRACTICE for Examples 1 and 2 1. How many steps do you save in the proof in Example 1 by using the Right Angles Congruence Theorem? ANSWER 2 Steps 2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.

GUIDED PRACTICE for Examples 1 and 2 Write a proof. Given: 1 and 3 are complements; 3 and 5 are complements. Prove: ∠ 1 5 ANSWER

GUIDED PRACTICE for Examples 1 and 2 Statements (Reasons) 1. 1 and 3 are complements; 3 and 5 are complements. (Given) 2. ∠ 1 5 Congruent Complements Theorem.