2.6 What you should learn Why you should learn it PROVING STATEMENTS ABOUT ANGLES What you should learn GOAL 1 Justify statements about congruent angles. GOAL 2 Prove properties about special pairs of angles Why you should learn it Properties of special pairs of angles help you determine angles in real-life applications, such as design work.

2.6 PROVING STATEMENTS ABOUT ANGLES 1 GOAL PROPERTIES OF CONGRUENT ANGLES VOCABULARY PROPERTIES OF ANGLE CONGRUENCE Reflexive Symmetric Transitive EXAMPLE 1

Extra Example 1 Given: Prove: A C B 1 2 3 4 Statements Reasons 1. 1. 1. 1. 2. 2. 3. 3. 4. 4. Given Transitive Prop. of  Given Transitive Prop. of  EXAMPLE 2

Extra Example 2 Given: Prove: 1 2 4 3 Statements Reasons 1. 1. Given 2. 2. 3. 3. 4. 4. 5. 5. Transitive Prop. of  Def. of  Given Subs. Prop. of =

All right angles are congruent. CONGRUENCE THEOREM All right angles are congruent. EXAMPLE 3

Extra Example 3 Given: Prove: A B C D Statements Reasons 1. 1. Given 2. 2. 3. 3. 4. 4. Transitive Prop. of 

Checkpoint Given: Prove: A B C D E F Statements Reasons 1. 1. Given 2. 2. 3. 3. 4. 4. Transitive Prop. of 

2.6 PROVING STATEMENTS ABOUT ANGLES 2 GOAL USING CONGRUENCE OF ANGLES CONGRUENT SUPPLEMENTS THEOREM Two angles supplementary to the same angle (or congruent angles) are congruent CONGRUENT COMPLEMENTS THEOREM Two angles complementary to the same angle (or congruent angles) are congruent In proofs, these may be abbreviated as  Supp. Thm. and  Comp. Thm. EXAMPLE 4

Extra Example 4 Given: Prove: 1 2 3 4 Statements Reasons 1. 1. Given 2. 2. 3. 3. 4. 4. Transitive Prop. of =  Complements Thm.

If two angles form a linear pair, then they are supplementary. Checkpoint 1. In a diagram, are supplementary and are supplementary. Explain how to show that Using the definition of supplementary angles, So by the transitive property of equality. So by the subtraction property of equality. Therefore, by the definition of congruent angles. LINEAR PAIR POSTULATE EXAMPLE 5 If two angles form a linear pair, then they are supplementary.

Extra Example 5 In the diagram is right. Explain how to show A B C D E F 1 2 3 4 Using the substitution property, you know that by the Angle Addition Postulate. The diagram shows that Substitute 150° for to show VERTICAL ANGLES THEOREM EXAMPLE 6 Vertical angles are congruent.

Extra Example 6 Given: are a linear pair, are a linear pair. Prove: 1 2 3 Statements Reasons 1. 1. Given 2. 2. 3. 3. Linear Pair Post.  Supplements Thm.

Checkpoint 1. Find the measures of the angles in the diagram given and are complementary and 1 2 78° 3 4

QUESTIONS?