Geometry Section 4-2C Organizing a Proof Pg. 266 Be ready to grade 4-2B

Proof: A sequence of true statements placed in a logical order. Types of statements to include: The given information. Information that can be assumed from the figure. Definitions Page 839 contains a list of postulates and theorems by chapter. Postulates Algebraic properties Theorems that have already been proven. Every statement you put in your proof must be a result of something from above it.

3 types of proofs: Paragraph form:

3 types of proofs: Two-column form:

3 types of proofs: Flow-proof form:

Explore: Important: Mark the illustration as the reasoning progresses. 1 E C A B D 2 5 6 3 4 Given: Ð1 @ Ð4 AC @ CD Ð5 @ Ð6 Prove: rABC @ rDEC Ð1 and Ð2 form a linear pair, as do Ð3 and Ð4. Thus Ð1 is supplementary to Ð2, and Ð3 is supplementary to Ð4 because the angles in a linear pair The tells us that Ð1 @ Ð4. Therefore, Ð2 @ Ð3 because are supplementary. given information supplements of congruent angles are congruent. We also know that AC @ CD, and Ð5 @ Ð 6 from given information. Therefore, we can conclude that rABC @ rDEC by the Postulate. ASA

Try It: Important: Mark the illustration as the reasoning progresses. 1 W X Z V Y Given: X is the midpoint of VZ Ð1 @ Ð2 Prove: rVXW @ rZXY 2 Statements Reasons Scrambled Reasons 1. X is the midpoint of VZ. 1. a. Supplements of @ Ð’s are @ 2. VX @ XZ 2. b. Given 3. ÐWXV @ ÐYXZ 3. c. SAA 4. Ð1 @ Ð2 4. d. Def. of midpoint 5. rVXW @ rZXY 5. e. Vert. Ð’s are @. f. Right Ð’s are @. b. Given b. Given c. SAA d. Def. of midpoint e. Vert. Ð’s are @.

The given information tells us that AB @ XY Z Y A C B Given: rABC and rXYZ are right triangles with right angles ÐA and ÐX. AB @ XY ÐB @ ÐY Prove: rABC @ rXYZ Statements: Reasons AB @ XY Given 1 ÐA and ÐX are rt. angles Given 2 ÐA @ ÐX Rt. Angles are @ 3 ÐB @ ÐY Given 4 rABC @ rXYZ ASA The given information tells us that AB @ XY Therefore, rABC @ rXYZ by the ASA Postulate. ÐA @ ÐX because all right angles are congruent. And that ÐA and ÐX are right angles. We are also given that ÐB @ ÐY

Given: F is the midpoint of DH and EG. Prove: rDFE @ rHFG Statements: Reasons F is midpt. of DH and EG Given 1 DF @ HF Def. of midpoint 2 EF @ GF Def. of midpoint 3 ÐHFG @ ÐDFE Vert. angles are @ 4 rDFE @ rHFG SAS

Given: PQ || RS PQ @ RS Prove: rPQS @ rRSQ Statements: Reasons PQ @ RS Given 1 SQ @ SQ Reflexive 2 PQ || RS Given 3 ÐPQS @ ÐRSQ Alt. Int. are @ 4 rPQS @ rRSQ SAS

Homework: Practice 4-2C Skip the last one.