Parallelism “translates” into relations between angles as soon as a transversal cuts the two lines. It is these relations that make parallelism such a powerful tool for proofs.
Theorem — Angles formed by parallel lines and a transversal
If two lines and are parallel and a transversal cuts them, then:
- the alternate interior angles are congruent;
- the alternate exterior angles are congruent;
- the co-interior angles are supplementary;
- the corresponding angles are congruent.
The converse theorems also hold: if any one of these conditions is satisfied, then the lines are parallel.
Links
Topics: Euclidean geometry
Concepts: Alternate interior angles · Line · Parallel lines · Converse theorem · Transversal
Skills: Synthetic geometry