Angle Pairs & Parallel Lines Cut by a Transversal
Complementary, supplementary, and vertical angles — then the four famous transversal pairs and the parallel-line theorems that make them powerful.
1 Key ideas
Three relationships for two intersecting lines, then four pairs for a transversal cutting two lines.
Complementary angles
Two angles whose measures add to 90°.
35° + 55° = 90° — think “corner of a square”.
Supplementary angles
Two angles whose measures add to 180° — a straight line.
115° + 65° = 180° — a linear pair is always supplementary.
Vertical angles
Opposite angles where two lines cross. Always congruent — never call them “adjacent”.
Opposite = equal. Adjacent angles on a straight line are supplementary.
2 A transversal cuts two lines
A transversal is a line crossing two other lines, making 8 angles. Learn the four named pairs by position.
The 8-angle diagram
| Pair type | Where they sit | Example | If lines are parallel… |
|---|---|---|---|
| Corresponding | Same relative corner at each intersection (both upper-right, …) | ∠2 and ∠6 | Congruent |
| Alternate interior | Between the lines, on opposite sides of the transversal | ∠3 and ∠6 | Congruent |
| Alternate exterior | Outside the lines, on opposite sides of the transversal | ∠1 and ∠8 | Congruent |
| Same-side interior | Between the lines, on the same side of the transversal | ∠3 and ∠5 | Supplementary (add to 180°) |
3 Worked examples
Watch the relationship do the work — then solve for x.
Example 1 — parallel lines, alternate interior
- The two horizontal lines are parallel, so alternate interior angles are congruent.
- m∠3 = 65°. ∠6 is alternate interior to ∠3.
- Therefore m∠6 = 65°.
Example 2 — parallel lines, same-side interior
- Parallel lines: same-side interior angles are supplementary.
- m∠4 = 72°. ∠5 is same-side interior to ∠4.
- m∠5 = 180° − 72° = 108°.
Example 3 — algebra with vertical angles
- Vertical angles are congruent: 2x + 10 = 3x − 5.
- Subtract 2x from both sides: 10 = x − 5.
- Add 5: x = 15. Check: both angles = 2(15) + 10 = 40°. Positive — valid.
Always check that your answer makes every angle measure positive — a negative angle measure means something went wrong.
Example 4 — the converse: are the lines parallel?
- ∠2 and ∠6 are corresponding angles, both 80°.
- Converse theorem: if corresponding angles are congruent, the lines are parallel.
- Answer: yes — the lines must be parallel.