In Parallel Lines Cut by a Transversal you saw that parallel lines cut by a transversal make pairs of equal angles. Now you will prove those facts with the tools from Writing a Proof, and also prove their converses, which run the other way.
Vertical angles
Vertical Angles Theorem: vertical angles are congruent. Proof: When two lines cross, $\angle 1$ and $\angle 2$ form a linear pair, so $m\angle 1 + m\angle 2 = 180^\circ$. Also $\angle 2$ and $\angle 3$ form a linear pair, so $m\angle 2 + m\angle 3 = 180^\circ$. By substitution $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$, and subtracting $m\angle 2$ gives $m\angle 1 = m\angle 3$.
Parallel lines
Why it works
| Pair | Where the angles sit | With parallel lines |
|---|---|---|
| Corresponding | same position at each crossing | congruent |
| Alternate interior | between the lines, opposite sides of the transversal | congruent |
| Alternate exterior | outside the lines, opposite sides | congruent |
| Same-side interior | between the lines, same side | supplementary |
Worked example: Proving the Alternate Interior Angles Theorem
- 1. $\ell \parallel m$ — Given.
- 2. $\angle 2 \cong \angle 6$ — Corresponding Angles Postulate (both are above their line, right of $t$).
- 3. $\angle 2 \cong \angle 3$ — Vertical Angles Theorem.
- 4. $\angle 3 \cong \angle 6$ — Transitive Property of Congruence.
Answer: Alternate interior angles are congruent.
Worked example: Using a converse
- If alternate interior angles are congruent, the lines are parallel (the converse). So set them equal: $5x - 12 = 3x + 20$.
- $2x = 32$, so $x = 16$. Then both angles measure $68^\circ$.
Answer: $x = 16$ makes $\ell \parallel m$.
Worked example: An angle chase
- $\angle 1$ and $\angle 5$ are corresponding, so $m\angle 5 = 118^\circ$.
- $\angle 5$ and $\angle 6$ form a linear pair: $m\angle 6 = 180 - 118 = 62^\circ$.
Answer: $62^\circ$. The angles $\angle 1$ and $\angle 6$ are not one of the named pairs, so the chase took two steps.
Key idea
Common mistake