1. Proving Angles Congruent Chapter 2: Reasoning and Proof1 Objectives 1 To prove and apply theorems about angles

2. Proving Angles Congruent Chapter 2: Reasoning and Proof2 Key Concepts A statement that you prove true is a ____________. A paragraph proof is written as sentences in a paragraph. A __________ is a convincing argument that uses deductive reasoning. “Given”: lists what you know from the hypothesis of the theorem “Prove”: the conclusion of the theorem Diagram: records the given information visually

3. Proving Angles Congruent Chapter 2: Reasoning and Proof3 Key Concepts Given:  1 and  2 are vertical angles Prove:  1   2 Proof: By the Angle Addition Postulate, m  1 + m  3 = 180 and m  2 + m  3 = 180. By substitution, m  1 + m  3 = m  2 + m  3. Subtract m  3 from each side. You get m  1 = m  2, or  1   2. Theorem: Vertical angles are congruent.

4. Proving Angles Congruent Chapter 2: Reasoning and Proof4 Find the value of x.

5. Proving Angles Congruent Chapter 2: Reasoning and Proof5 Key Concepts Given:  1 and  2 are supplementary  3 and  2 are supplementary Prove:  1   3 Proof: By the definition of supplementary angles, m  1 + m  2 = 180 and m  3 + m  2 = 180. By substitution, m  1 + m  2 = m  3 + m  2. Subtract m  2 from each side. You get m  1 = m  3, or  1   3. Theorem: If two angles are supplements of the same angle, then the two angles are congruent.

6. Proving Angles Congruent Chapter 2: Reasoning and Proof6 Key Concepts Given:  1 and  2 are supplementary  3 and  4 are supplementary  2   4 Prove:  1   3 Proof: By the definition of supplementary angles, m  1 + m  2 = 180 and m  3 + m  4 = 180. By substitution, m  1 + m  2 = m  3 + m  4. Since  2   4, by the definition of congruence m  2 = m  4. By substitution m  1 + m  4 = m  3 + m  4. Subtract m  4 from each side. You get m  1 = m  3, or  1   3. Theorem: If two angles are supplements of congruent angles, then the two angles are congruent.

7. Proving Angles Congruent Chapter 2: Reasoning and Proof7 Key Concepts Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem: All right angles are congruent. Theorem: If two angles are congruent and supplementary, then each is a right angle.

8. Proving Angles Congruent Chapter 2: Reasoning and Proof8 Write a paragraph proof using the given, what you are to prove, and the diagram. Given: WX = YZ Prove: WY = XZ ●● ●●