Lines and Angles
91. Parallel Lines & Transversals · alternate, corresponding & co-interior angles
Alternate angles stay equal, corresponding angles stay equal, and co-interior angles always add to 180°.
When two parallel lines are cut by a transversal, three angle relations always hold: alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°. These follow from the parallelism by construction.
What this lesson covers
Try to break it
What happens if I make the transversal vertical? Or almost flat? Do the angle pairs still match up?
How you build it
Draw two parallel lines cut by a transversal.
- Draw line AB.
- Draw line CD parallel to AB.
- Draw transversal PQ intersecting AB and CD.
- Mark point I1 where the transversal PQ crosses line AB.
- Mark point I2 where the transversal PQ crosses line CD.
The proof, step by step
Prove that a transversal across parallel lines makes alternate and corresponding angles equal.
- Given: AB || CD and PQ is a transversal.
- ∠1 = ∠5 (Corresponding angles are equal when lines are parallel).
- ∠1 = ∠3 (Vertically opposite angles are equal).
- Therefore, ∠3 = ∠5 (Alternate interior angles are equal).
- ∠3 + ∠6 = 180° (Co-interior angles are supplementary).
Worked example
In the figure, lines l and m are parallel and t is a transversal. If ∠1 = 75°, find the measures of ∠3, ∠5, and ∠6.
Since ∠1 and ∠3 are vertically opposite, ∠3 = 75°. Since ∠1 and ∠5 are corresponding angles, ∠5 = 75°. Since ∠5 and ∠6 are co-interior angles, ∠6 = 180° - 75° = 105°.
- ∠3=75°, ∠5=75°, ∠6=105° — correct
- ∠3=105°, ∠5=75°, ∠6=75°
- ∠3=75°, ∠5=105°, ∠6=105°
- ∠3=105°, ∠5=105°, ∠6=75°