Parallelograms Theorem 2

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Statement

If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.

Given:

ABCD is a quadrilateral in which diagonals AC and BD

intersect at O such that AO=OC and BO=OD.

To prove:

ABCD is a parallelogram.

Proof:

In triangles AOB and COD,

AO = CO (given)

BO = OD (given)

(vertically opposite angles are equal)

D AOB D COD (SAS congruency condition)

Since these are alternate angles made by the transversal AC intersecting AB and CD.

AB||CD

Similarly, AD||BC

Hence ABCD is a parallelogram.