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 pointsAdjacent Angles Examples Adjacent angles: ∢ 1 and ∢ 2 are adjacent angles NOT adjacent angles: ∢ 3 and ∢ ABC are not adjacent angles

These angles are adjacent. 55º 35º 50º130º 80º 45º 85º 20º

These angles are NOT adjacent. 45º55º 50º 100º 35º

Tell whether the numbered angles are adjacent or nonadjacent. 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. a. b.c. Example 1:

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

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 complementary adjacent complementary nonadjacent Complementary Angles

Angle Pair Relationships Supplementary Angles – two angles whose measures have a sum of 180ºSupplementary Angles Examples: ∢ 3 and ∢ 4 are supplementary; ∢ P and ∢ Q are supplementary

supplementary nonadjacent supplementary adjacent 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. Supplementary Angles

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.

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 2:

Determine whether the angles are complementary, supplementary, or neither. a. b. 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. Example 3:

ANSWER supplementary ANSWER complementary Determine whether the angles are complementary, supplementary, or neither. 1. ANSWER neither Your Turn:

Find the angle measure. SOLUTION m  C = 90˚ – m  A = 90˚ – 47˚ = 43˚ Given that  A is a complement of  C and m  A = 47˚, find m  C. Example 4:

Find the angle measure. SOLUTION m  P = 180˚ – m  R = 180 ˚ – 36˚ = 144˚ Given that  P is a supplement of  R and m  R = 36˚, find m  P. Example 5:

 B is a complement of  D, and m  D = 79°. Find m  B  G is a supplement of  H, and m  G = 115°. Find m  H. ANSWER 11° ANSWER 65° Your Turn:

 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, But m  Z = 5( m  W ), Because 6 ( m  W ) = 90˚, so m  W + 5( m  W) = 90˚. you know that m  W = 15˚. m  W + m  Z = 90˚. Example 6:

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 =

Supplement Theorem Theorem 2.2 (Supplement Theorem) If two angles form a linear pair, then they are supplementary angles. m ∠1 + m∠2 =

SOLUTION  7 and  9 are both supplementary to  8. So, by the Congruent supplements Theorem,  7   9.  7 and  8 are supplementary, and  8 and  9 are supplementary. Name a pair of congruent angles. Explain your reasoning. Example 7:

ANSWER  10   12;  10 and  12 are both complementary to  11, so  10   12 by the Congruent Complements Theorem. In the diagram, m  10 + m  11 = 90°, and m  11 + m  12 = 90°. Name a pair of congruent angles. Explain your reasoning. Your Turn:

Practice Time!

Directions: Identify each pair of angles as supplementary, complementary, or neither.

#1 60º 120º

#1 60º 120º Supplementary Angles

#2 60º 30º

#2 60º 30º Complementary Angles

#3 60º 40º

#3 60º 40º neither

#4 45º135º

#4 45º135º Supplementary Angles

#5 65º 25º

#5 65º 25º Complementary Angles

#6 50º 90º

#6 50º 90º neither

Directions: Determine the missing angle.

#1 45º?º?º

#1 45º135º

#2 65º ?º?º

#2 65º 25º

#3 50º ?º?º

#3 50º 130º

#4 40º ?º?º

#4 40º 50º

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 : #1-37 odd, odd