Alternate Exterior Angles Theorem

In order to prove that alternate exterior angles are congruent, it will be shown that ∠ 1 and ∠ 2 are congruent.

Notice that by definition, ∠ 2 and ∠ 8 are corresponding angles. Therefore, by the Corresponding Angles Theorem, they are congruent angles. ∠ 2 ≅ ∠ 8 Furthermore, by definition, ∠ 8 and ∠ 1 are vertical angles. Therefore, by the Vertical Angles Theorem, ∠ 8 and ∠ 1 are congruent angles. ∠ 8 ≅ ∠ 1 Then, by applying the Transitive Property of Congruence, ∠ 2 and ∠ 1 can be concluded to be congruent angles as well. ∠ 2 ≅ ∠ 8 ∠ 8 ≅ ∠ 1 ⇒ ∠ 2 ≅ ∠ 1 The same reasoning applies to the other pair of alternate exterior angles. Therefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate exterior angles are congruent.

Two-Column Proof

The previous proof can be summarized in the following two-column table.

Consider the points of intersection as well as two more points on each line.

Next, A, B, C, and D will be translated in the direction of the transversal so that points A, C, and D lie on l_2. Then, A, B, and C will be rotated 180^(∘) about F.

After this combination of rigid motions, A, B, C, and D are mapped onto A', B', C', and D'. This means that ∠ ADB is mapped onto ∠ A'D'B'. Therefore, ∠ ADB and ∠ A'D'B' are congruent angles. ∠ ADB ≅ ∠ A'D'B' Since D' and F share the same location, A' lies on FG and B' lies on FH. Because of this, ∠ A'D'B' is congruent to ∠ GFH. ∠ A'D'B' ≅ ∠ GFH Applying the Transitive Property of Congruence, ∠ ADB is congruent to ∠ GFH. ∠ ADB ≅ ∠ A'D'B' ∠ A'D'B' ≅ ∠ GFH ⇓ ∠ ADB ≅ ∠ GFH It has been proved that one pair of alternate exterior angles is congruent. Further, since C' lies on EF, it can also be proven that ∠ C'D'B' is congruent to ∠ EFH. ∠ C'D'B' ≅ ∠ EFH Applying the Transitive Property of Congruence again, ∠ CDB is congruent to ∠ EFH. ∠ CDB ≅ ∠ C'D'B' ∠ C'D'B' ≅ ∠ EFH ⇓ ∠ CDB ≅ ∠ EFH To conclude, it has been obtained that both pairs of alternate exterior angles are congruent.