if the diagonals of a quadrilateral bisect each other then it is a parallelogram

if the diagonals of a quadrilateral bisect each other, then it is a parallelogram", then the answer is given as,

Given: A quadrilateral ABCD in which diagonals AC and BD intersect at O such that OA = OC and OB = OD.

To prove: ABCD is a parallelogram.

Proof: 

In ΔAOD and ΔBOC,

OA = OC  (Given)

OD = OB  (Given)

∠AOD = ∠BOC  (Vertically opposite angles)

∴ ΔAOD ΔBOC  (SAS congruence criterion)

⇒ ∠OAD = ∠OCB  (CPCT)

∴ AD || BC    ...(1)  (If a transversal intersect two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel)

Similarly, AB || CD    ...(2)

From (1) and (2), we have

AB || CD and AD || BC

Hence, ABCD is a parallelogram  (A quadrilateral is a parallel, if both pair of its opposite sides is parallel)

Cheers!