A Parallelogram, whose Diagonals are of Equal Length, is a Rectangle

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)

9th Grade Math

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