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

Explanation

ABCD is a quadrilateral with AC and BD are diagonals intersecting at O
It is given that diagonals bisect each other.
∴OA=OC andbOB=OD
Step - 2: Proving that the quadrilateral is a parallelogram.
In ΔAOD and ΔCOB,
OA=OC
∠AOD=∠COB...(Vertically opposite angles)
OD=OB
Therefore ΔAOD≅ΔCOB by SAS Congruence rule.
∴∠OAD=∠OCB
Similarly, we can prove that ΔAOB≅ΔCOD
∴∠ABO=∠CDO
So, for lines AB and BC with transversal BD, we can write
∠ABO and ∠CDO are alternate angles and are equal i.e. AB||CD.
Similarly, for lines AD and BC, with transversal AC, ∠OAD and ∠OCB are alternate angles and are equal. Therefore, BC||AD.
Thus, in ABCD, both pair of opposite sides are parallel.
Therefore, ABCD is a parallelogram.
Hence, Proved that quadrilateral whose diagonals bisect each other is a parallelogram.