Question

PQRS is a parallelogram. If it's diagnals are equal, then find the value of angle SPQ​

Answer 1

Answer:

Given: PQRS is a parallelogram in which PQ ∥ SR, PS ∥ QR and diagonal PR ⊥diagonal QS.

To prove: PQRS is a square, i.e., PQ = QR = RS = SP and an angle, say ∠SPQ = 90°.

Proof:

In ∆PQR and ∆RSP,

∠QPR = ∠PRS (Since, PQ ∥ SR and QR is a transversal)

∠QRP = ∠SPR (Since, QR ∥ PS and PR is a transversal)

PR = PR (Common Side). 

Therefore, ∆PQR ≅ ∆RSP (By AAS criterion of congruency).

Therefore, PQ = SR (CPCTC).

Similarly, ∆PQS ≅ ∆RSQ (By AAS criterion of congruency).

Therefore, PS = QR (CPCTC).

∆OPQ ≅ ∆ORS (By AAS criterion of congruency).

Therefore, OP = OR (CPCTC).

Similarly, ∆POQ ≅ ∆ROQ (By SAS criterion of congruency).

Therefore, PQ = QR (CPCTC).

Therefore, PQ = QR = RS = SP. (Proved)

∆SPQ ≅ ∆RQP (By SSS criterion of congruency).

Therefore, ∠SPQ = ∠RQP (CPCTC).

But ∠SPQ + ∠RQP = 180° (Since, PS ∥ QR).

Therefore, ∠SPQ = ∠RQP = \(\frac{180°}{2}\) = 90°. (Proved).