Question

Consider parallel lines cut by a transversal.

Parallel lines q and s are cut by transversal r. On line q where it intersects line r, 4 angles are created. Labeled clockwise, from uppercase left: angle 1, angle 2, angle 4, angle 3. On line s where it intersects line r, 4 angles are created. Labeled clockwise, from uppercase left: angle 5, angle 6, angle 8, angle 7.
Explain which theorems, definitions, or combinations of both can be used to prove that alternate exterior angles are congruent.

Answer 1

You can prove the congruence of alternate exterior angles using the Corresponding Angles Theorem (or postulate), the Vertical Angles Theorem, and the transitive property of congruence. Here is a short labelled proof for the pair (∠1, ∠8); the other pair (∠2, ∠7) is analogous.

Given: q ∥ s and r is a transversal.

1. By the Corresponding Angles Theorem (parallel lines cut by a transversal → corresponding angles are congruent), ∠1 ≅ ∠5 because ∠1 and ∠5 occupy corresponding positions at the two intersections.
2. ∠5 and ∠8 are a pair of vertical angles at the intersection of r and s, so by the Vertical Angles Theorem, ∠5 ≅ ∠8.
3. By transitivity of congruence (if ∠1 ≅ ∠5 and ∠5 ≅ ∠8, then ∠1 ≅ ∠8), we conclude ∠1 ≅ ∠8.

Thus alternate exterior angles (∠1 and ∠8, and similarly ∠2 and ∠7) are congruent.

Theorems/definitions used: definition of parallel lines, Corresponding Angles Theorem (parallel ⇒ corresponding angles congruent), Vertical Angles Theorem, and transitive property of congruence.