Proving Lines Parallel Lesson 12 Proving Lines Parallel

Write the converse of the conditional statements & determine their truth value If an angle is a right angle, then it measures 90º. If two angles are vertical angles, then they are congruent. If an angle measures 90º, then it is a right angle. True If two angles are congruent, then they are vertical angles. False

What is the converse of Postulate 11 If two parallel lines are cut by a transversal, then the corresponding angles are congruent. If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. What do you think the truth value is? The converse of a postulate or theorem is not necessarily true When the converse is true, then it is stated in a separate postulate or theorem Today’s lesson will take the converses from in Investigation 1 to prove lines are parallel

Postulate 12: Converse of the Corresponding Angles Postulate If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. If ∠1 ≅ ∠2, then 𝑤 𝑢 .

𝑚∠3=34°, prove 𝑚 𝑛 𝑚∠7 = 34° by vertical angle theorem ∠3 ≅ ∠7 by def. of congruent angles Therefore 𝑚 𝑛 by the converse of corresponding angles postulate (conv. corr. ∠’s) This is the proof for our next theorem about alternate interior angles

Theorem 12-1: Converse of the Alternate Interior Angles Theorem If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. If ∠3 ≅ ∠6, then 𝑤 𝑢 .

Prove 𝑚 𝑛 , given ∠2 ≅ ∠7 Given ∠ 2 ≅ ∠ 7 ∠6 ≅ ∠7 by the Vert. Angle Thm ∠2 ≅ ∠6 by transitive property So 𝑚 𝑛 by Converse of Corr. Angles Since ∠2 & ∠7 are alternate exterior angles, this is the proof for our next theorem

Theorem 12-2: Converse of the Alternate Exterior Angles Theorem If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel. If ∠2 ≅ ∠7, then 𝑤 𝑢 .

𝑚∠6=52°, prove 𝑚 𝑛 𝑚∠2 + 128 = 180 by Linear Pair Theorem 𝑚∠2 = 52° and 𝑚∠6 = 52°, therefore by def. ≅ ∠’s ∠2 ≅ ∠6 Therefore by conv. corr. ∠’s 𝑚 𝑛 Since 𝑚∠6 + 128° = 180° this is the proof for our next theorem about same-side interior angles

Theorem 12-3: Converse of the Same-Side Interior Angles Theorem If two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel. If 𝑚∠3 + 𝑚∠5 = 180°, then 𝑤 𝑢 .

Theorem 12-4: Converse of the Same-Side Exterior Angles Theorem If two lines are cut by a transversal and the same-side exterior angles are supplementary, then the lines are parallel. If 𝑚∠2 + 𝑚∠8 = 180°, then 𝑤 𝑢 .

Questions/Review When using these or any other conditional statement, keep in mind you must have the true hypothesis before you can determine the is true conclusion Also, remember that not all converses are true like the ones we studied today If they are true, then they most likely have a post/thm of their own What information do we need to conclude that lines p & q are parallel or not?