Section shows how to prove two lines to be parallel using the converses of the postulates and theorems of previous lessons

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

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

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

8. Sailing
If two boats sail at an angle of 45o 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. Practice
• TB p153: 10-28 even, 44-48 (all)
Maybe you could be a bit more explicit at step 3.