1. Proving Lines Parallel
Lesson 12
Proving Lines Parallel

2. 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

3. 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

4. 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 .

5. 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

6. 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 .

7. 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

8. 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 .

9. 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∠ ° = 180° this is the proof for our next theorem about same-side interior angles

10. 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 .

11. 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 .

12. 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?