Theorems About Parallelograms
Rule

Converse Parallelogram Diagonals Theorem

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Quadrilateral with diagonals that bisect each other

Based on the diagram above, the following relation holds true.

If and bisect each other, then is a parallelogram.

Proof

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

Quadrilateral with diagonals that bisect each other

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

Quadrilateral with diagonals that bisect each other

Applying a similar reasoning, it can be concluded that and are congruent triangles. Consequently, and are also congruent.

Quadrilateral with diagonals that bisect each other

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

Quadrilateral with diagonals that bisect each other
Exercises