Bell Work 1) Solve for each variable 2) Solve for each variable 3 and 4) Transitive Property of equality Definition of Congruence Given Definition of Congruence

Outcomes I will be able to: 1) Use angle congruence properties 2) Prove properties about special angle relationships

Properties of Angle Congruence Reflexive Property Symmetric Property Transitive Property For any Angle A, If, then If, then

Proof Practice Prove the Transitive Property of Congruence for angles Given: Statements 1. ∠ A ≅ ∠ B and ∠ B ≅∠ C 2. m ∠ A = m ∠ B 3. m ∠ B = m ∠ C 4. m ∠ A = m ∠ C 5. ∠ A ≅ ∠ C Prove: Reasons 1. Given 2. Definition of Congruent Angles 3. Definition of Congruent Angles 4. Transitive/Substitution 5. Definition of Congruent Angles

Use the Transitive Property Given: m ∠ 3 = 40 °, ∠1 ≅ ∠2, ∠2 ≅ ∠3 Statements 1. m∠3 = 40°, ∠1 ≅ ∠2, ∠2 ≅ ∠3 2. ∠1 ≅ ∠3 3. m ∠1 = m∠3 4. m∠1 = 40° Prove: Angle 1 = 40° Reasons 1. Given 2. Transitive Property of Congruence. 3. Def. of Congruence 4. Substitution

Angle Investigation 1) Fold your paper in half. 2) Place the corner of your second piece of paper at the vertex of right angles and trace it 3) Label the four angles from left to right 1, 2, 3, 4 as shown in the picture 4) Answer the following questions on the sheet of paper.

Complementary Questions

Angle Relationship Theorems Right Angle Congruence Theorem: All right angles are congruent Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. Ex: Angle 1 is supplementary to Angle 2 and Angle 3 is supplementary to Angle 2 Therefore, Congruent Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then they are congruent. Ex: Angle 1 and Angle 3 are both complementary to Angle 2. Therefore, Angle 1 is congruent to Angle 3

Proof Practice Prove the Congruent Complements Theorem Statements 1.Angle 1 and Angle 2 are complements; Angle 3 and Angle 4 are complements; Angle 2 is congruent to Angle Reasons 1. Given 2. Definition of Complementary Angles 3. Transitive Property of Equality 4. Def. of Congruent Angles 5. Substitution Prop. of Equality 6. Subtraction Prop. of Equality 7. Def. of Congruent Angles

Angle Relationship Theorems Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. Meaning: Angle 1 + Angle 2 = 180° Vertical Angles Theorem: Vertical angles are congruent Meaning:

Example Using Angle Relationships By definition of a right angle, Angle 3 = 90°. Angle 2 and Angle 5 are vertical angles and Angle 5 = 57°, so Angle 2 = 57°. Angle 1 and Angle 5 form a linear pair, so Angle 1 + Angle 5 = 180°. When you substitute 57° for angle 5, and solve for Angle 1, the result is Angle 1 = 123°. Angle 4 and Angle 5 are complementary, so Angle 4 + Angle 5 = 90°. When you substitute 57° for Angle 5 and solve for Angle 4, the result is Angle 4 = 33°

Check Point ON YOUR OWN Using your knowledge of angle pair relationships, solve for each angle in the diagram. Then, match the diagram in column A with the appropriate theorem/postulate in column B

Exit Quiz Given the following information, complete the two-column proof below