Objectives: Use a transversal in proving lines parallel

1. Section 3-2 Proving Lines ParallelTPI 32C: use inductive and deductive reasoning to make conjectures, draw conclusions, and solve problemsTPI 32E: write and defend indirect and direct proofs • Objectives: • Use a transversal in proving lines parallel • Relate parallel and perpendicular lines Recall Vocabulary Conditional Statement: If p  q. (p = hypothesis q = conclusion) Converse of a conditional: If q  p. (switch the hypothesis and conclusion) Biconditional: p iff q; can write only if both conditional and converse are true.

2. Using a Transversal and Proving the Converse of Theorems Prove theorems based on their converses Corresponding Angles Postulate from Section 3-1: If a transversal intersects parallel lines, then corresponding angles are congruent. Converse of the Corresponding Angles Postulate If two lines and a transversal form corresponding angles, then the two lines are parallel. We have done paragraph proofs and two-column proofs. Now we will look at a third type of proof: The Flow Proof

3. Proof of the Converse of the Alternate Interior Angles Theorem Theorem 3-3: Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Transitive Property of Congruence: If A B and B  C, then A  C. Converse of Corresponding Angles Theorem: If two lines and a transversal form corresponding angles, then the two lines are parallel. Flow Proof: arrows show logical connections between statements reasons are written below the statements Given: 1  2 Prove: ℓ||m 1  2 Given 3  2 ℓ||m 1  3 Transitive Prop. of Congruence If corresponding s are , then the lines are ||. Vertical s are 

4. Proof of the Converse of the Same-side Interior Angles Theorem Theorem 3-4: Converse of the Same-side Interior Angles Theorem If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. Converse of Corresponding Angles Theorem: If two lines and a transversal form corresponding angles, then the two lines are parallel. Congruent Supplements Thm: If two angles are supplements of the same angle (or congruent angles), then the two angles are congruent. Given: 1 and 2 are supplementary Prove:   m 1 & 2 are suppl. 2  3   m Given 1 & 3 are suppl. Converse of Corresponding s Postulate  supplement Thm (p. 98) Def of supplementary angles

5. Using Theorem 3-4 (Converse of Same-side Interior Angles) Which lines, if any, must be parallel if 1  2? Justify your answer with a theorem or postulate. What type of angles are 1 and 2? Alternate Interior Angles: We know they are congruent. Which Theorem states: If alternate interior angles are congruent, then the lines are parallel. Converse of the Alternate Interior Angles Theorem Conclude that ray DE is parallel to ray KC.

6. Relate Parallel and Perpendicular Lines Theorem 3-5: If two lines are parallel to the same line, then they are parallel to each other. This theorem guarentees that the lines you draw are parallel.

7. 1 and 2 are corresponding angles by the converse of the corresponding angles theorem. r || s Theorem 3-6: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m||n Proof of Theorem 3-6: Given: r  t and s  t Prove: r || s Write a paragraph proof: Study what is given, the diagram, and what you need to prove. 1 and 2 are right angles by def. of  so they are congruent.

8. Relate Algebra and Geometry Do Now Find the value of x for which ℓ||m. Justify each step. 2x + 6 = 40 Corr. Angles are  SPE 2x = 34 DPE x = 17

9. Real-World Connection: Drafting Do Now An artist uses a drafting tool in the diagram at the right. The artist draws a line, slides the triangle along the flat surface, and draws another line. Explain why the drawn lines must be parallel. The corresponding angles are congruent, so the lines are parallel by the Converse of Corresponding Angles Postulate. If two lines and a transversal form corr. s, then the two lines are parallel.