The document discusses several theorems and properties for proving when two lines are parallel, including: 1) If corresponding angles formed by a transversal intersecting two lines are congruent, then the lines are parallel. 2) If alternate interior angles or alternate exterior angles formed by a transversal are congruent, then the lines are parallel. 3) If consecutive interior angles formed by a transversal are supplementary, then the lines are parallel. It also provides an example of using these properties to prove that if two boats sail at 45 degree angles to the constant wind, their paths will not cross since their paths are parallel.

1. Proving Lines are Parallel

2. Properties of Parallel Lines Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Converse: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

3. Biconditional: Two lines cut by a transversal are parallel if and only if they the corresponding angles are congruent.

4. If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Theorem: Alternate Interior Angles: Converse: If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

5. If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Theorem: Consecutive Interior Angles: Converse: If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

6. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem: 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. Proof of Alternate Interior Angles Converse Statement Reason 1 ∠ 1 ≅ ∠ 2 Given 2 ∠ 2 ≅ ∠ 3 Vertical angles theorem 3 ∠ 1 ≅ ∠ 3 Transitive property of congruence 4 l ⊥ m Converse of corresponding angles postulate

8. Sailing If two boats sail at an angle of 45 o to the wind and the wind is constant, will their paths ever cross?

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

10. Homework Exercise 3.4 page 153: 1-37, odd.