1. Pre-AP Bellwork
1) Solve for p.
(3p – 5)°

2. 3-2 Proving Lines Parallel

3. Postulate 3-2: Converse of the Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

4. 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 lines are parallel.

5. Theorem 3-4: Converse of the Sam-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.

6. Theorem 3.10: Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

7. Prove the Alternate Interior Angles Converse
Given: 1  2
Prove: m ║ n 3 m 2 1 n

8. Example 1: Proof of Alternate Interior Converse
Statements:
1  2
2  3
1  3
m ║ n
Reasons:
Given
Vertical Angles
Transitive prop.
Corresponding angles converse

9. Proof of the Consecutive Interior Angles Converse
Given: 4 and 5 are supplementary
Prove: g ║ h g 6 5 4 h

10. Paragraph Proof
You are given that 4 and 5 are supplementary. By the Linear Pair Postulate, 5 and 6 are also supplementary because they form a linear pair. By the Congruent Supplements Theorem, it follows that 4  6. Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

11. Find the value of x that makes j ║ k.
Solution:
Lines j and k will be parallel if the marked angles are supplementary.
x + 4x = 180 
5x = 180 
X = 36 
4x = 144 
So, if x = 36, then j ║ k.
4x x

12. Using Parallel Converses: Using Corresponding Angles Converse
SAILING. If two boats sail at a 45 angle to the wind as shown, and the wind is constant, will their paths ever cross? Explain

13. Solution:
Because corresponding angles are congruent, the boats’ paths are parallel. Parallel lines do not intersect, so the boats’ paths will not cross.

14. Example 5: Identifying parallel lines
Decide which rays are parallel. H E G
58
61
62
59 C A B D
A. Is EB parallel to HD?
B. Is EA parallel to HC?

15. Example 5: Identifying parallel lines
Decide which rays are parallel. H E G
58
61 B D
Is EB parallel to HD?
mBEH = 58
m DHG = 61 The angles are corresponding, but not congruent, so EB and HD are not parallel.

16. Example 5: Identifying parallel lines
Decide which rays are parallel. H E G
120
120 C A
B. Is EA parallel to HC?
m AEH = 62 + 58
m CHG = 59 + 61
AEH and CHG are congruent corresponding angles, so EA ║HC.

17. Conclusion:
Two lines are cut by a transversal. How can you prove the lines are parallel?
Show that either a pair of alternate interior angles, or a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.