Here we will discuss about a quadrilateral is a parallelogram if its diagonals bisect each other.
Given: PQRS is a quadrilateral whose diagonals PR and QS bisect each other at O, i.e., OP = OR and OQ = OS.
To prove: PQRS is a parallelogram.
Proof: In ∆OPQ and ∆ORS,
OP = OR, OQ = OS (Given);
∠POQ = ∠ROS (Opposite angles).
Therefore, ∆OPQ ≅ ∆ORS (by SAS criterion of congruency)
Therefore, ∠OPQ = ∠ORS (CPCTC).
So, PQ ∥ SR (From equal alternate angles).
Similarly, from ∆OQR and ∆OSP we get PS ∥ QR.
Therefore, PQRS is a parallelogram.