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 $\angle AEB$ and $\angle CED$ 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, $\overline{AB}$ and $\overline{CD}$ are congruent.

Applying a similar reasoning, it can be concluded that $△AED$ and $△CEB$ are congruent triangles. Consequently, $\overline{AD}$ and $\overline{BC}$ 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.