Geometric Theorems Deductive Reasoning Proofs

1. Warm Up Complete each sentence. 1.If the measures of two angles are ? , then the two angles are congruent. 2. If two angles form a ? , then they are supplementary. 3. If two angles are complementary to the same angle, then the two angles are ? . equal linear pair congruent

2. Learning Targets I will prove geometric theorems by using deductive reasoning.

4. Example Use the given information to write a two-column proof. Given: 2 and 3 are complementary 1  3 Prove: 2and 1are complementary

5. Example 1 Continued Two-column proof: 1. 2 and 3 are complementary 1  3 1. Given 2. m2 + m3 = 90° 2. Def. complementary angles 3. m1 = m3 3. Def. congruent angles 4. m2 + m1 = 90° 4. Substitution property 5. 2 and 1 are complementary 5. Def. complementary angles

6. Example Use the given information to write a two-column proof. Given: 2  4 Prove: m1  m3

7. Example Continued

8. Example Use the given paragraph proof to write a two-column proof. Given: m1 + m2 = m4 Prove: m3 + m1 + m2 = 180° Paragraph Proof: It is given that m1 + m2 = m4. 3 and 4 are supplementary by the Linear Pair Theorem. So m3 + m4 = 180° by definition. By Substitution, m3 + m1 + m2 = 180°.

9. Example Continued Two-column proof: 1. Given 1. m1 + m2 = m4 2. 3 and 4 are supplementary 2. Linear Pair Theorem 3. m3 + m4 = 180° 3. Def. supplementary angles 4. m3 + m1 + m2 = 180° 4. Substitution Property

10. Example Use the given information to write a two-column proof. Given: WXYis a right angle. 1  3 Prove: 1 and 2 are complementary.

11. Example Continued

12. m3 + m4 = 90° 3 and 4 are comp. Example Provide the reasons for the two-column proof shown below. Given: 1 and 2 are complementary Prove: 3 and 4 are complementary

13. Example Use the information to write a two-column proof. Given: 1  4 Prove: 2  3

14. Example Continued

15. Homework Page 124, #9 – 16.