Math 444 Quiz #2

Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Answer

Let ABCD be a quadrilateral with diagonals AC and BD intersecting at point O. We are given that O is the midpoint of both AC and of BD. This means that OA = OC and OB = OD.

By vertical angles, angle AOB = angle COD. Thus by SAS triangle AOB is congruent to triangle COD.

By corresponding angles in the congruent triangles, angle CAB = angle OAB is congruent to angle OCD = angle ACD. Thus the transversal AC meets the lines AB and CD with congruent alternate interior angles. This means the lines AB and CD are parallel.

Thus the quadrilateral ABCD has opposite sides AB and CD parallel. Since no special choices were made, it is also true that the other pair of opposite sides is also parallel.

Thus by definition, ABCD is a parallelogram.