1. Lesson 2-3 Proving Theorems (page 43) Essential Question Can you justify the conclusion of a conditional statement?

2. Proving Theorems What is a theorem? Statements that can be proved.

3. If M is the midpoint of, then AM = ____ AB and MB = ____ AB. Theorem 2-1 ½ Midpoint Theorem Given: Prove: ½ A M B

4. StatementsReasons 1.___________________ ___________________ 2.___________________ ___________________ 3.___________________ ___________________ 4.______________________ ___________________ 5.___________________ ___________________ 6.___________________ ___________________ Given: Prove: A M B See page 43 for the proof.

5. Definition of Midpoint Example:Given R is the midpoint of. Give the reason that justifies each statement. S R Q P (a) _________________________________

6. (b) _________________________________ Midpoint Theorem Example:Given R is the midpoint of. Give the reason that justifies each statement. S R Q P

7. (c) _________________________________ Segment Addition Post. Example:Given R is the midpoint of. Give the reason that justifies each statement. S R Q P

8. (d) _________________________________ Def. of Segment Bisector Example:Given R is the midpoint of. Give the reason that justifies each statement. S R Q P

9. If is the bisector of ∠ ABC, then m ∠ ABX = ____ m ∠ ABC and m ∠ XBC = ____ m ∠ ABC. Theorem 2-2 ½ Angle Bisector Th m Given: Prove: ½ B A X C

10. StatementsReasons 1.___________________ ___________________ 2.___________________ ___________________ ___________________ 3.___________________ ___________________ 4.______________________ ___________________ ___________________ 5.___________________ ___________________ 6.___________________ ___________________ Given: Prove: B A X C See page 45 Classroom Exercises #10.

11. (a) m ∠ CFD = ½ m ∠ CFE _________________________________ Angle Bisector Theorem Example:Given bisects ∠ CFE. Give the reason that justifies each statement. C D EF

12. (b) m ∠ CFD = m ∠ DFE _________________________________ Def. of Angle Bisector Example:Given bisects ∠ CFE. Give the reason that justifies each statement. C D EF

13. (c) CD + DE = CE _________________________________ Segment Addition Post. Example:Given bisects ∠ CFE. Give the reason that justifies each statement. C D EF

14. Reason Used in Proofs 1.Given information 2.Definitions 3.Postulates (including properties from algebra) 4.Theorems - only ones that have already been proved!

15. Deductive Reasoning: … proving statements by reasoning from accepted postulates, definitions, theorems, and given information. Example: 2-column proofs

16. Note: Definitions can be written as biconditionals (combine conditional and converse), i.e. the conditional and converse are both true.

17. Example of a Biconditional: Conditional: If an angle is a right angle, then its measure is 90º. Converse: If the measure of an angle is 90º, then the angle is a right angle. Biconditional: An angle is a right angle if and only if its measure is 90º.

18. StatementsReasons 1.______________________________________ ___________________ ________________ 2.______________________ ___________________ 3.______________________ ___________________ 4.______________________________________ 5.______________________________________ Given: M is the midpt. of N is the midpt. of PQ = RS Prove: PM = RN M is the midpt. of Multiplication Prop. Midpoint Theorem Substitution Prop. P MQ PM = RN N is the midpt. of ½ PQ = ½ RS PM = ½ PQ R N S PQ = RS Given Midpoint Theorem RN = ½ RS

19. Assignment Written Exercises on page 46 DO NOW: 9 to 12 ALL numbers GRADED: 1 to 8 ALL numbers Assignment Worksheet on Lesson 2-3 Prepare for a Quiz on Lessons 2-1 to 2-3: Using Deductive Reasoning Can you justify the conclusion of a conditional statement?