Step | Description | Justification |
1 | a || b | This is a criterion, a necessary condition. |
2 | ∠1 ≅ ∠5 | The Parallel Line Postulate states that, given a transversal of parallel lines, corresponding angles are congruent. |
3 | ∠5 ≅ ∠7 | The Vertical Angle Theorem states that vertical angles are congruent. |
4 | ∠1 ≅ ∠7 | The Transitivity of Congruence of Angles Theorem states that if two angles are congruent to the same angle, they are congruent to each other. In this case since ∠1 ≅ ∠5 and ∠5 ≅ ∠7, then ∠1 ≅ ∠7. |
5 | ∠2 ≅ ∠8 Q.E.D. | A similar argument can be made to show that ∠2 ≅ ∠8. |
Table 1: Proof of the Alternate Interior Angles Theorem |