Proving Angles Congruent During this lesson, you will: Determine and apply conjectures about angle relationships Prove and apply theorems about angles Mrs. McConaughy Geometry

Part I: Discovering Angle Relationships Mrs. McConaughy Geometry

Definitions: Special Angle Pairs complementary angles Two angles are ___________________ if their measures add up to 90. Two angles are ___________________ if their measures add up to 180. supplementary angles Mrs. McConaughy Geometry

Vocabulary Review: Pairs of Angles Formed By Intersecting Lines Opposite (non-adjacent) angles, formed by intersecting lines, which share a common vertex and whose sides are opposite rays are called ______________. Adjacent angles formed by intersecting lines which share a common vertex, a common side, and with one side formed by opposite rays are called ____________. vertical angles linear pairs Mrs. McConaughy Geometry

Given the following diagram, identify all vertical angle pairs: 1 2 4 3 ∠ 1 & ∠ 3 ∠ 2 & ∠ 4 Mrs. McConaughy Geometry

Given the following diagram, identify all linear pairs of angles: 2 4 8 6 ∠ 6 & ∠ 8 ∠ 8 & ∠ 2 ∠ 4 & ∠ 6 ∠ 2 & ∠ 4 Mrs. McConaughy Geometry

Investigative Results: If two angles are vertical angles, then the angles are _________. (VERTICAL ANGLES CONJECTURE) If two angles are a linear pair of angles, then the angles are ______________ (____). (LINEAR PAIR CONJECTURE) congruent supplementary 180 Mrs. McConaughy Geometry

If two angles are equal and supplementary, what must be true of the two angles? If two angles are both equal in measure and supplementary, then each angle measures ____. (EQUAL SUPPLEMENTS CONJECTURE) 90 Mrs. McConaughy Geometry

Examples: Use your conjectures to find the measure of each lettered angle. Example A a b c Example B a b c 70 30 Mrs. McConaughy Geometry

Examples: Use your conjectures to a. find the value of the variable. EXAMPLE C (3y + 20) (5y – 16) EXAMPLE D (2x – 6) Vertical Angles Are Congruent Linear Pairs Are Supplementary (3x + 31) 5y – 16 = 3y + 20 3x + 31 + 2x – 6 = 180 5y = 3y + 36 5y = 3y + 36 5x + 25 = 180 2y = 36 5x = 155 Mrs. McConaughy Geometry y = 18 x = 31

Homework Assignment: Discovering Angle Relationships WS 1-5 all, plus select problems from text. Mrs. McConaughy Geometry

Part 2: Proving and Applying Theorems About Angles Mrs. McConaughy Geometry

Congruent Supplements Theorem If two angles are supplements of congruent angles, then the two angles are congruent. Mrs. McConaughy Geometry

Given: ∠A supp ∠B; ∠C supp ∠D; ∠B  ∠C Prove: ∠A  ∠D STATEMENT REASON 1. 2. 3. 4. 5. 6. 7. ∠A supp ∠B; ∠C supp ∠D Given. m ∠A + m ∠B = 180; m ∠C + m ∠D = 180 Def. of supp. ∠’s m ∠A + m ∠B = m ∠C + m ∠D . Substitution Prop. of = ∠B  ∠C Given. Def. of  m ∠B = m ∠C m ∠A = m ∠ D Subtraction Prop. of = ∠A  ∠C Def. of  Mrs. McConaughy Geometry

Vocabulary: Corollary A _________ of a theorem is a theorem whose proof contains only a few additional statements in addition to the original proof. EXAMPLE: If two angles are supplements of the same angle, then the two angles are congruent. Mrs. McConaughy Geometry

Congruent Complements Theorem If two angles are complements of congruent angles, then the two angles are congruent. COROLLARY: If two angles are complements of the same angle, then the two angles are congruent. Mrs. McConaughy Geometry

Given: ∠A comp . ∠B; ∠C comp. ∠D; ∠B  ∠C Prove: ∠A  ∠D STATEMENT REASON 1. 2. 3. 4. 5. 6. 6 7. ∠A comp ∠B; ∠C comp ∠D Given. m ∠A + m ∠B = 90; m ∠C + m ∠D = 90 Def. of supp. ∠’s m ∠A + m ∠B = m ∠C + m ∠D Substitution Prop. of = ∠B  ∠C Given. m ∠B = m ∠C Def. of  m ∠A = m ∠ D (-) Prop. of = ∠A  ∠C Def. of  Mrs. McConaughy Geometry

Vertical angles are congruent. Theorem Vertical angles are congruent. Given: ∠ 1 and ∠ 3 are vertical angles 1 3 2 Prove: ∠ 1  ∠ 3 Mrs. McConaughy Geometry

Given: ∠ 1 and ∠ 3 are vertical angles Prove: ∠ 1  ∠ 3 STATEMENT REASON 1. 2. 3. 4. 5. ∠ 1 and ∠ 3 are vertical angles Given. Def. of linear pair ∠ 1 and ∠2 are a linear pair ∠ 2 and ∠3 are a linear pair Def. of linear pair ∠1 supp ∠2; ∠3 supp ∠2 Linear pairs are supp. ∠ 1  ∠ 3 Supp. of same ∠  Mrs. McConaughy Geometry

All right angles are congruent. Theorem All right angles are congruent. Mrs. McConaughy Geometry

Final Checks for Understanding In the following exercises, ∠ 1 and ∠ 3 are a linear pair, ∠ 1 and ∠ 4 are a linear pair, and ∠ 1 and ∠ 2 are vertical angles. Is the statement true? ∠ 1  ∠ 3 b. ∠ 1  ∠ 2 c. ∠ 1  ∠ 4 d. ∠ 3  ∠ 2 e. ∠ 3  ∠ 4 f. m∠ 2 + m ∠ 3 = 180 Mrs. McConaughy Geometry

Homework Assignment Pages 100-101: 10-18 all. 32-35 all. Prove: 19 & 35 all. Mrs. McConaughy Geometry