1. Section 3-2 Proving Lines Parallel TPI 32C: use inductive and deductive reasoning to make conjectures, draw conclusions, and solve problems TPI 32E: write and defend indirect and direct proofs Objectives: Use a transversal in proving lines parallel Relate parallel and perpendicular lines 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 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 Flow Proof: arrows show logical connections between statements reasons are written below the statements If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Theorem 3-3: Converse of the Alternate Interior Angles Theorem Given:  1   2 Prove: ℓ ||m  1   2 Given  1   3 Vertical  s are   3   2 Transitive Prop. of Congruence ℓ || m If corresponding  s are , then the lines are ||. Converse of Corresponding Angles Theorem: If two lines and a transversal form corresponding angles, then the two lines are parallel. Transitive Property of Congruence: If  A  B and  B   C, then  A   C.

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

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 If two lines are parallel to the same line, then they are parallel to each other. Theorem 3-5:

7. 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.  1 and  2 are corresponding angles by the converse of the corresponding angles theorem. r || s

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

9. Real-World Connection: Drafting 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.