Parallelogram - Proof That Diagonals Bisect Each Other

Proof That Diagonals Bisect Each Other

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:

(alternate interior angles are equal in measure)

(alternate interior angles are equal in measure).

(since these are angles that a transversal makes with parallel lines AB and DC).

Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.

Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).

Therefore,

Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.

Separately, since the diagonals AC and BD bisect each other at point E, point E is the midpoint of each diagonal.