Final answer:
The Same-Side Interior Angles Theorem does not prove two angles to be congruent but states that such angles are supplementary. In contrast, Vertical Angles Theorem, Corresponding Angles Postulate, and Alternate Interior Angles Theorem all establish congruency between pairs of angles under certain conditions.
Step-by-step explanation:
Among the options given—Vertical Angles Theorem, Corresponding Angles Postulate, Alternate Interior Angles Theorem, and Same-Side Interior Angles Theorem—the one that does not prove two angles to be congruent is the Same-Side Interior Angles Theorem.
The Same-Side Interior Angles Theorem actually states that if two parallel lines are cut by a transversal, then the same-side interior angles are supplementary, not congruent. This means that the two angles add up to 180 degrees, indicating that they sum up to a straight line but do not necessarily have to be equal themselves.
On the other hand, the Vertical Angles Theorem states that vertical angles, which are the angles opposite each other when two lines intersect, are congruent. The Corresponding Angles Postulate indicates that if two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent. Lastly, the Alternate Interior Angles Theorem states that if two parallel lines are intersected by a transversal, then each pair of alternate interior angles are congruent.