In this post, we show that the diagonals of a parallelogram bisect each other. We will use the fact that the diagonal of a parallelogram is a transversal to the opposite sides of the parallelogram. The transversal and the sides form congruent alternate interior angles. Recall that the alternate interior angles formed by a transversal and parallel lines are congruent.

**Given**

Parallelogram with diagonals and .

**What to Show**

and .

**Proof**

because they are alternate interior angles of parallel lines that pass through and cut by the transversal (**A**).

because opposite sides of a parallelogram are congruent (**S**).

because they are alternate interior angles of parallel lines that pass through and cut by the transversal (**A**).

By **ASA **Triangle Congruence theorem, .

and because corresponding sides of congruent triangles are congruent.