The setup
When a transversal crosses two parallel lines, it creates eight angles: four at each intersection. These angles come in predictable pairs.
Label the angles at the top intersection 1, 2, 3, 4 (top-left, top-right, bottom-left, bottom-right) and at the bottom intersection 5, 6, 7, 8 in the same positions.
The angle pairs
| Pair | Where | Rule (parallel lines) | Example pair |
|---|---|---|---|
| Vertically opposite angles | across an intersection | equal (always) | 1 and 4 |
| Corresponding angles | same position at each intersection ("F" shape) | equal | 1 and 5 |
| Alternate interior angles | between the lines, opposite sides ("Z" shape) | equal | 3 and 6 |
| Alternate exterior angles | outside the lines, opposite sides | equal | 1 and 8 |
| Co-interior angles | between the lines, same side ("C" or "U" shape) | add to 180° | 3 and 5 |
| Angles on a straight line | next to each other | add to 180° | 1 and 2 |
Spotting the shapes
- F shape → corresponding (equal)
- Z shape → alternate (equal)
- C or U shape → co-interior (supplementary)
Two sizes only
With parallel lines, all eight angles have only two sizes. If one angle is 65°, every angle is either 65° or 115°.
Example
A transversal crosses parallel lines. One angle is 70°.
- Its vertically opposite angle: 70°
- Its neighbour on the straight line: 110°
- Its corresponding angle: 70°
- Its co-interior partner: 110°
Using algebra
Example: Alternate interior angles are (3x + 10)° and 85°.
- 3x + 10 = 85
- 3x = 75
- x = 25
Example: Co-interior angles are (2y)° and (y + 30)°.
- 2y + y + 30 = 180
- 3y = 150
- y = 50
Proving lines are parallel
The rules work backwards: if corresponding angles are equal (or alternate angles are equal, or co-interior angles add to 180°), then the lines are parallel.
Worked examples
Example 1: One angle is 48°. Find its corresponding angle.
48°.
Example 2: One angle is 48°. Find its co-interior angle.
180 − 48 = 132°.
Example 3: Corresponding angles are 4x° and 100°. Find x.
4x = 100 → x = 25.
Common mistakes
- Using these rules when the lines aren’t parallel. Only vertically opposite angles are always equal.
- Confusing alternate and co-interior angles.
- Assuming co-interior angles are equal. They add to 180°.
Try it yourself
- Vertically opposite angles: one is 132°. The other?
- Alternate interior angles: one is 57°. The other?
- Co-interior angles: one is 57°. The other?
- Corresponding angles are (5x − 20)° and 80°. Find x.
Answers: 1) 132° 2) 57° 3) 123° 4) 20
Quick recap
- A transversal across parallel lines creates angle pairs.
- Corresponding (F) and alternate (Z) angles are equal.
- Co-interior (C) angles add to 180°.