2.8 Proving Angle Relationships

1. 2.8 Proving Angle Relationships What you’ll learn: To write proofs involving supplementary and complementary angles. To write proofs involving congruent and right angles.

2. Postulates Postulate 2.10 – Protractor Postulate – angles can be measured using a protractor. Postulate 2.11 – Angle Addition Postulate If R is in the interior of PQS, then mPQR+mRQS=mPQS If mPQR+mRQS=mPQS, then R is in the interior of PQS. P R Q S

3. Theorems 2.3 Supplement Theorem – If 2 s form a linear pair, then they are supplementary s. 2.4 Complement Theorem – if the noncommon sides of 2 adjacent s form a right , then the s are complementary s. 2.5. Congruence of s is reflexive, symmetric, and transitive. Reflexive: AA Symmetric: If AB, then BA Transitive: If AB and BC, then AC. 2.6 Angles supplementary to the same angle or to congruent angles are congruent.

4. More Theorems 2.7 Angles complementary to the same angle or to congruent angles are congruent. 2.8 If 2 angles are vertical angles, then they are congruent. 2.9 Perpendicular lines intersect to form 4 right angles. 2.10 All right angles are congruent. 2.11 Perpendicular lines form congruent adjacent angles 2-12 If 2 s are  and supplementary, then each  is a right . 2-13 if 2  s form a linear pair, then they are right s.

5. Don’t forget Definition of supplementary angles – if 2 angles are supplementary, they add to be 180. Definition of complementary angles – if 2 angles are complementary, they add to be 90. Definition of congruent angles: if 2 angles are congruent, they are equal in measure and vice-versa. Definition of angle bisector – an angle bisector creates 2 congruent angles. Definition of right angles – if an angle is right, its measure is 90

6. Find the measure of each numbered angle. 1. 1=65, 2=? 2. 1=32, 2=?, 3=? 3. 1=125, 2=?, 3=?, 4=? 4. 4=x-32, 2=175-2x 2 1 2 1 2 4 1 3 3 1 2 4 3

7. Sometimes, always, never • Supplementary angles are congruent. sometimes • If 2 angles form a linear pair, then they are complementary. never • Two vertical angles are supplementary. sometimes • Two angles that are congruent to the same angle are congruent to each other. always

8. Write a 2-column proof.Given: ABC is a right angle.Prove: 1 and 2 are complementary angles A 1 C 2 B • ABC is a right angle. • mABC =90 • mABC =m1+m2 • 90=m1+m2 • 1 and 2 are complementary angles Given defn. right angles Angle addition postulate Substitution Defn of comp. angles

9. Homeworkp. 11216-32 even, 38,46-54 even