Question

If the diagonals PR and QS of a quadriateral PQRS bisect each other, then prove that PQRS is a parallelogram.​

Answer 1

Proof: In ∆OPQ and ∆ORS,

Proof: In ∆OPQ and ∆ORS,OP = OR, OQ = OS (Given);

Proof: In ∆OPQ and ∆ORS,OP = OR, OQ = OS (Given);∠POQ = ∠ROS (Opposite angles).

Proof: In ∆OPQ and ∆ORS,OP = OR, OQ = OS (Given);∠POQ = ∠ROS (Opposite angles).Therefore, ∆OPQ ≅ ∆ORS (by SAS criterion of congruency)

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).

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).

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.

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.