Properties of Parallel Lines

In the photograph, the vapor trail of the high-flying aircraft suggests a transversal of the parallel trails of the low-flying aircraft.

Image shows a photograph of two flying aircraft with their vapor trails.

The same-size angles that appear to be formed by the vapor trails suggest the postulate and theorems below.

Key Concepts

Postulate 3-1 Corresponding Angles Postulate

If a transversal intersects two parallel lines, then corresponding angles are congruent.

1 2

Key Concepts

Theorem 3-1 Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

1 3

Theorem 3-2 Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

m1 + m2 = 180

You can display the steps that prove a theorem in a two-column proof.

Reading Math

These symbols indicate lines a and b are parallel.

Two-Column Proof of Theorem 3-1

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

Given a || b
Prove 1 3

To write a proof, you may find it helpful to first write a plan for the proof. In a plan, you write key statements that connect what you prove to what is given.

Planning a Proof

Developing Proof For Theorem 3-2 below, study what is given, what you are to prove, and the diagram. Then write a plan for a proof.

If two lines are parallel and cut by a transversal, then same-side interior angles are supplementary.

Given a || b
Prove 1 and 2 are supplementary.
Plan To prove that m1 + m2 = 180, show that m3 + m2 = 180. Then show that m1 = m3 and substitute m1 for m3.

Check Understanding

When you see two parallel lines and a transversal, and you know the measure of one angle, you can find the measures of all the angles. This is illustrated in Example 4.