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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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?