A Quadrilateral is a Parallelogram if its Diagonals Bisect each Other

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.

9th Grade Math

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