Here we will prove that a parallelogram, whose diagonals are of equal length, is a rectangle.
Given: PQRS is a parallelogram in which PQ ∥ SR, PS ∥ QR and PR = QS.
To prove: PQRS is a parallelogram, i.e., in the parallelogram PQRS, one angle, say ∠QPS = 90°.
Proof:
In ∆PQR and ∆RSP,
∠RPQ = ∠PRS (Since, SR ∥ PQ),
∠QRP = ∠SPR (Since, PS ∥ QR),
PR = PR
Therefore, ∆PQR ≅ ∆RSP, (By AAS criterion of congruency)
Therefore, QR = PS (CPCTC).
In ∆PQR and ∆QPS,
QR = PS
PR = QS (Given),
PQ = PQ.
Therefore, ∆PQR ≅ ∆QPS (By SSS criterion of congruency)
∠PQR = ∠QPS (CPCTC).
But ∠PQR + ∠QPS = 180° (Since, QR ∥ PS)
Therefore, ∠PQR = ∠QPS = 90° (Proved)
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