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!