Converse Parallelogram Diagonals Theorem

Let E be point of intersection of the diagonals of a quadrilateral. Since the diagonals bisect each other, E is the midpoint of each diagonal.

Because ∠ AEB and ∠ CED are vertical angles, they are congruent by the Vertical Angles Theorem. Therefore, by the Side-Angle-Side Congruence Theorem, △ AEB and △ CED are congruent triangles. Since corresponding parts of congruent figures are congruent, AB and CD are congruent.

Applying a similar reasoning, it can be concluded that △ AED and △ CEB are congruent triangles. Consequently, AD and BC are also congruent.

Finally, since both pairs of opposite sides of quadrilateral ABCD are congruent, the Converse Parallelogram Opposite Sides Theorem states that ABCD is a parallelogram.