Proving Lines are Parallel

Properties of Parallel LinesCorresponding 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.

Biconditional:

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

• 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.

• 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.

• 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.

Proof of Alternate Interior Angles Converse

Statement Reason

1 ∠1 ≅ ∠2 Given

2 ∠2 ≅ ∠3 Vertical angles theorem

3 ∠1 ≅ ∠3Transitive property of congruence

4 l ⊥ mConverse of corresponding angles postulate

Sailing

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

Solution

• Because corresponding angles are congruent, the boats’ paths are parallel.

• Parallel lines do not intersect, so the boats’ paths will not cross.

Homework

• Exercise 3.4 page 153: 1-37, odd.