Lines and Angles
92. The Parallel Promise · equal alternate angles mean parallel lines
When alternate angles are equal, the lines are parallel.
Two lines cut by a transversal are parallel if any of these hold: alternate angles equal, corresponding angles equal, OR co-interior angles supplementary (sum to 180°). Any one of these conditions implies the other two.
What this lesson covers
Try to break it
Notice how the alternate angles change as you drag. When they are equal, the lines are parallel. When they differ, the lines tilt away from each other.
How you build it
Draw a transversal across two parallel lines and compare the alternate angles.
- Draw line l.
- Draw transversal t crossing l.
- Mark point A where t crosses l.
- Mark point B on the transversal.
- Through B, draw line m parallel to line l.
The proof, step by step
Prove that the two lines are parallel when their alternate angles are equal.
- Let lines l and m be cut by transversal t at A and B.
- Given alternate angles ∠1 = ∠2.
- Vertically opposite angle to ∠2 is ∠3, so ∠2 = ∠3.
- Thus ∠1 = ∠3 (corresponding angles).
- Since corresponding angles are equal, l || m.
Worked example
Two lines l and m are cut by a transversal t. If the alternate interior angles are (2x + 10)° and (3x - 20)°, find x for which l || m.
For l || m, alternate interior angles must be equal. So, 2x + 10 = 3x - 20. Solving gives x = 30.
- 20
- 30 — correct
- 40
- 50