If two lines are cut by a transversal so that a pair of consecutive interior angles (same side interior angles) are congruent, then the lines are parallel.

This statement is actually false. If two lines are cut by a transversal so that a pair of consecutive interior angles are congruent, then the lines are not necessarily parallel.

Here's why:

1. When two lines are cut by a transversal, it forms eight angles. Among these, there are two pairs of consecutive interior angles (also known as same side interior angles).

2. According to the Consecutive Interior Angles Theorem, if two lines are cut by a transversal to form a pair of congruent consecutive interior angles, then the lines are parallel. However, the theorem actually states that if the lines are parallel, then the consecutive interior angles are supplementary (add up to 180 degrees), not necessarily congruent (equal).

3. Therefore, if a pair of consecutive interior angles are congruent (meaning they are equal), then they each measure 90 degrees. This is a special case where the lines are perpendicular, not parallel.

4. So