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Transcript
Complementary and Supplementary Angles Section 2.3 Objective • Find measures of complementary and supplementary angles. Key Vocabulary • • • • Complementary angles Supplementary angles Adjacent angles Theorem Theorems • 2.1 Congruent Complements Theorem • 2.2 Congruent Supplements Theorem Pairs of Angles • Adjacent Angles – two angles that lie in the same plane, have a common vertex and a common side, but no common interior points • Examples • Adjacent angles: ∢1 and ∢2 are adjacent angles • NOT adjacent angles: ∢3 and ∢ABC are not adjacent angles These angles are adjacent. 80º 45º 35º 55º 130º 85º 20º 50º These angles are NOT adjacent. 100º 50º 35º 35º 55º 45º Example 1: Tell whether the numbered angles are adjacent or nonadjacent. b. a. c. SOLUTION a. Because the angles do not share a common vertex or side, 1 and 2 are nonadjacent. b. Because the angles share a common vertex and side, and they do not have any common interior points, 3 and 4 are adjacent. c. Although 5 and 6 share a common vertex, they do not share a common side. Therefore, 5 and 6 are nonadjacent. Angle Pair Relationships • Two Types • Complementary Angles • Supplementary Angles • Remember, angle measures are real numbers, so the operations for real numbers and algebra can apply to angles. Angle Pair Relationships • Complementary Angles – two angles whose measures have a sum of 90º • Examples: ∢1 and ∢2 are complementary; ∢A is complementary to ∢B Complementary Angles Two angles are complementary angles if the sum of their measurements is 90˚. Each angle is the complement of the other. Complementary angles can be adjacent or nonadjacent. 4 1 3 2 complementary adjacent complementary nonadjacent Angle Pair Relationships • Supplementary Angles – two angles whose measures have a sum of 180º • Examples: ∢3 and ∢4 are supplementary; ∢P and ∢Q are supplementary Supplementary Angles Two angles are supplementary angles if the sum of their measurements is 180˚. Each angle is the supplement of the other. Supplementary angles can be adjacent or nonadjacent. 7 5 8 6 supplementary adjacent supplementary nonadjacent Identifying Complementary and Supplementary Angles • Complementary angles make a Corner of a piece of paper. • Supplementary angles make up the Sides of a piece of paper. Example 2: State whether the two angles are complementary, supplementary, or neither. SOLUTION The angle showing 4:00 has a measure of 120˚ and the angle showing 10:00 has a measure of 60˚. Because the sum of these two measures is 180˚, the angles are supplementary. Example 3: Determine whether the angles are complementary, supplementary, or neither. b. a. c. SOLUTION a. Because 22° + 158° = 180°, the angles are supplementary. b. Because 15° + 85° = 100°, the angles are neither complementary nor supplementary. c. Because 55° + 35° = 90°, the angles are complementary. Your Turn: Determine whether the angles are complementary, supplementary, or neither. 1. ANSWER neither ANSWER complementary ANSWER supplementary 2. 3. Example 4: Find the angle measure. Given that A is a complement of C and m A = 47˚, find mC. SOLUTION mC = 90˚ – m A = 90˚ – 47˚ = 43˚ Example 5: Find the angle measure. Given that P is a supplement of R and mR = 36˚, find mP. SOLUTION mP = 180˚ – mR = 180 ˚ – 36˚ = 144˚ Your Turn: 1. B is a complement of D, and mD = 79°. Find mB. ANSWER 11° 2. G is a supplement of H, and mG = 115°. Find mH. ANSWER 65° Example 6: W and Z are complementary. The measure of Z is 5 times the measure of W. Find m W SOLUTION Because the angles are complementary, m W + m Z = 90˚. But m Z = 5( m W ), so m W + 5( m W) = 90˚. Because 6(m W) = 90˚, you know that m W = 15˚. Theorems • We use undefined terms, definitions, postulates, and algebraic properties of equality to prove that other statements or conjectures are true. Once a statement or conjecture has been shown to be true, it is called a theorem. • Once proven true, a theorem can be used like a definition or postulate to justify other statements or conjectures. • Thus, a theorem is a true statement that follows from other true statements. Complement Theorem Theorem 2.1 (Complement Theorem) If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. m∠1 + m∠2 = 90 1 2 Supplement Theorem Theorem 2.2 (Supplement Theorem) If two angles form a linear pair, then they are supplementary angles. m∠1 + m∠2 = 180 1 2 Example 7: 7 and 8 are supplementary, and 8 and 9 are supplementary. Name a pair of congruent angles. Explain your reasoning. SOLUTION 7 and 9 are both supplementary to 8. So, by the Congruent supplements Theorem, 7 9. Your Turn: In the diagram, m10 + m11 = 90°, and m11 + m12 = 90°. Name a pair of congruent angles. Explain your reasoning. ANSWER 10 12; 10 and 12 are both complementary to 11, so 10 12 by the Congruent Complements Theorem. Practice Time! Directions: Identify each pair of angles as supplementary, complementary, or neither. #1 120º 60º #1 120º 60º Supplementary Angles #2 30º 60º #2 30º 60º Complementary Angles #3 40º 60º #3 60º 40º neither #4 135º 45º #4 135º 45º Supplementary Angles #5 25º 65º #5 25º 65º Complementary Angles #6 90º 50º #6 90º 50º neither Directions: Determine the missing angle. #1 ?º 45º #1 135º 45º #2 ?º 65º #2 25º 65º #3 ?º 50º #3 130º 50º #4 ?º 40º #4 50º 40º Joke Time • Why did the geometry student get so excited after they finished a jigsaw puzzle in only 6 months? • Because on the box it said from 2-4 years. • Why did the geometry student climb the chain-link fence? • To see what was on the other side. • How did the geometry student die drinking milk? • The cow fell on them. Assignment • Section 2-3, pg. 70-73: #1-37 odd, 41-53 odd