Final answer:
To prove the converse of the alternate exterior angles theorem, one shows that if two alternate exterior angles are congruent, then the lines are parallel using a structured two-column proof. The proof involves the definition of congruent angles, the linear pair postulate, and the Converse of the Same-Side Interior Angles Postulate.
Step-by-step explanation:
To prove the converse of the alternate exterior angles theorem, we must demonstrate that if two alternate exterior angles are congruent, then the lines are parallel. A two-column proof is a method that consists of two columns: one for statements and one for the corresponding reasons.
Two-Column Proof of Converse Alternate Exterior Angles Theorem
Assume that two lines are cut by a transversal and alternate exterior angles are congruent.
By the definition of congruent angles, the measures of the alternate exterior angles are equal.
According to the Converse of the Same-Side Interior Angles Postulate (if same-side interior angles are supplementary, then the lines are parallel), the same-side interior angles must also be supplementary because the sum of the measures of linear pairs is 180 degrees.
Since the alternate exterior angles are congruent, their corresponding same-side interior angles are supplementary.
Therefore, by the postulate mentioned above, the lines cut by the transversal must be parallel.
This completes the proof of the converse of the alternate exterior angles theorem, showing that if two alternate exterior angles are congruent, then the lines must be parallel.