Section 3-2: Proving Lines Parallel Goal: Be able to use a transversal in proving lines parallel. Warm up: Write the converse of each conditional statement. Determine the truth value of the converse. 1.) If a triangle is a right triangle, then it has a 90 degree angle. converse: If a triangle has a 90 degree angle, then it is a right triangle. truth value: TRUE 2.) If two angles are vertical angles, then they are congruent. converse: If two angles are congruent, then they are vertical angles. truth value: FALSE 3.) If two angles are same-side interior angles, then they are supplementary. converse: If two angles are supplementary, then they are same- side interior angles. truth value: FALSE

Converse of the Corresponding Angles Postulate If two lines and a transversal form ___________________ _________ that are _____________, then the two lines are _______________. 1 2 m n correspondingangles congruent parallel corresponding angles congruent  || If then m || n.

Converse of the Alternate Interior Angles Theorem If two lines and a transversal form ___________________ _________ that are _____________, then the two lines are _______________. 3 4 m n alternate interiorangles congruent parallel alternate interior angles congruent  || If then m || n.

Converse of the Alternate Exterior Angles Theorem If two lines and a transversal form ___________________ _________ that are _____________, then the two lines are _______________. 5 6 m n Alternate exteriorangles congruent parallel alternate exterior angles congruent  || If then m || n.

Converse of the Same-Side Interior Angles Theorem If two lines and a transversal form _____________ ___________________ that are ________________, then the two lines are _______________. 7 8 m n same-side interior angles supplementary parallel same-side interior angles supplementary  || If and are supplementary, then m || n.

Converse of the Same-Side Exterior Angles Theorem If two lines and a transversal form _____________ ___________________ that are ________________, then the two lines are _______________ m n same-side exterior angles supplementary parallel same-side exterior angles supplementary  || If and are supplementary, then m || n.

Ex 1: Complete the proof about the Converse of the Same-Side Interior Angles Theorem. Given: is supplementary to Prove: m || n StatementsReasons 1.) is supplementary to1.) 2.) is supplementary to2.) 3.) 4.) m || n4.) m n Given If two angles form a linear pair, then they are supplementary. If 2 angles are supplementary to the same angle, then they are congruent. corresponding angles  ||

Ex 2: Find the value of x for which m || n. (14 + 3x)° (5x – 66)° Ex 3: Find the value of x for which m || n. 93 ° (x + 44)° m n m n alternate interior angles congruent  || x = 5x – = 2x 40 = x alternate exterior angles congruent  || x + 44 = 93 x = 49

Ex 4: Using the given information, which lines, if any, can you conclude are parallel? Justify each conclusion with a theorem or postulate. a.) is supplementary b.) is supplementary c.) d.) e.) f.) rs p q p || q, same side interior angles r || s, same side interior angles p || q, corresponding angles none r || s, same side exterior angles