DWG NO. 03.2 — Lesson 2 of 4

Proving Lines Parallel

Unit 3: Parallel and Perpendicular Lines · ~15–30 min

Objective

Use the converse angle theorems to prove that two lines cut by a transversal are parallel.

Flipping the theorems around

Lesson 3.1 started from "the lines are parallel" and worked out what that says about the angles. This lesson runs the logic backwards: start from what the angles measure, and use that to prove the lines are parallel. Each theorem from 3.1 has a converse that does exactly this.

The marked angles at P and Q are alternate exterior angles. If they're congruent, j ∥ k.

The converse theorems

Converse of the Corresponding Angles Postulate: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
Converse of the Alternate Interior Angles Theorem: If alternate interior angles are congruent, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem: If alternate exterior angles are congruent, then the lines are parallel.
Converse of the Consecutive Interior Angles Theorem: If consecutive interior angles are supplementary, then the lines are parallel.

One more useful shortcut follows from these: if two lines are each perpendicular to the same third line, they must be parallel to each other — you'll lean on this again once slope enters the picture in Lesson 3.3.

Worked Example 1 · Identifying the right converse

ProblemIn the diagram above, the marked angles at P and Q are both 128°. Show that j ∥ k.

1.The two marked angles sit outside lines j and k, on opposite sides of transversal t — that makes them alternate exterior angles.

2.They're given as congruent (128° = 128°).

3.By the Converse of the Alternate Exterior Angles Theorem, congruent alternate exterior angles mean the lines are parallel.

j ∥ k, by the Converse of the Alternate Exterior Angles Theorem

Worked Example 2 · Solving for x, then proving parallel

ProblemLines j and k are cut by transversal t. One same-side interior angle measures (2x + 8)° and the other measures (4x + 22)°. Find the value of x that makes j ∥ k.

1.Same-side interior angles are consecutive interior angles. By the converse theorem, the lines are parallel exactly when these angles are supplementary.

2.Set the sum equal to 180: (2x + 8) + (4x + 22) = 180.

3.Combine and solve: 6x + 30 = 180, so 6x = 150, and x = 25.

x = 25 makes the consecutive interior angles supplementary (75° and 105°), so j ∥ k

Guided practice

1.Two lines are cut by a transversal, and a pair of corresponding angles both measure 87°. Are the lines parallel? Name the theorem.
Show answer
Yes. By the Converse of the Corresponding Angles Postulate, congruent corresponding angles mean the lines are parallel.
2.A pair of consecutive interior angles measure 100° and 82°. Are the two lines parallel? Explain.
Show answer
No. 100° + 82° = 182°, not 180°, so the angles aren't supplementary — the Converse of the Consecutive Interior Angles Theorem doesn't apply, and the lines are not parallel.
3.Two alternate interior angles are given as (3x + 5)° and (5x − 25)°. Find the value of x that makes the lines parallel.
Show answer
x = 15. Alternate interior angles must be congruent for the lines to be parallel: 3x + 5 = 5x − 25, so 30 = 2x, and x = 15.
4.Explain why "two lines cut by a transversal always form congruent corresponding angles" is a false statement on its own, without the extra converse condition.
Show answer
Corresponding angles are only guaranteed congruent when the lines are parallel. If the lines aren't parallel, corresponding angles can differ — congruence is the condition that proves parallelism, not an automatic result.
5.Line r is perpendicular to line s, and line r is also perpendicular to line q. What can you conclude about s and q?
Show answer
s ∥ q. Two lines perpendicular to the same line are parallel to each other.

← Lesson 3.1 Unit 3 Overview Lesson 3.3 →