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.
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)