Proving Lines Parallel

By the end of this lesson, you will be able to use the converses of the parallel line theorems to prove two lines are parallel when cut by a transversal, and write formal two-column proofs establishing that lines are parallel.

The Big Idea

In the previous lesson, we started with "lines are parallel" and derived angle relationships. Now we flip the logic: we start with angle relationships and prove the lines must be parallel. This is the converse of each theorem, and all converses are true — making these tests completely reversible.

The Converse Theorems

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

Converse of Alternate Interior Angles Theorem: If alternate interior angles are congruent, the lines are parallel.

Converse of Alternate Exterior Angles Theorem: If alternate exterior angles are congruent, the lines are parallel.

Converse of Consecutive Interior Angles Theorem: If consecutive interior angles are supplementary, the lines are parallel.

Example 1 — Identifying Parallel Lines

Lines $\ell$ and $m$ are cut by transversal $t$. Determine if $\ell \parallel m$:

(a) $\angle 3 = 68°$ and $\angle 5 = 68°$ (alternate interior angles) → parallel (converse of Alternate Interior Angles Theorem)

(b) $\angle 4 = 115°$ and $\angle 6 = 65°$ (co-interior angles, sum = 180°) → parallel (converse of Consecutive Interior Angles Theorem)

(c) $\angle 1 = 72°$ and $\angle 5 = 75°$ (corresponding, unequal) → NOT parallel

Example 2 — Two-Column Proof

Given: $\angle 1 \cong \angle 2$ (both labeled at their respective intersections as corresponding angles)

Prove: $\ell \parallel m$

Statement	Reason
$\angle 1 \cong \angle 2$	Given
$\ell \parallel m$	Converse of Corresponding Angles Postulate

Example 3 — Longer Proof

Given: $\angle ABD \cong \angle CDB$ (alternate interior angles formed by transversal $BD$ between lines $AB$ and $CD$)

Prove: $AB \parallel CD$

Statement