if the diagonals of a parallelogram are equal in length, show that the parallelogram is a rectangle
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Let PQRS be a parallelogram. To show that PQRS is a rectangle, we have to prove that one of its interior angles is 90º.
In ΔPQR and ΔSRQ,
PQ = SR (Opposite sides of a parallelogram are equal)
QR = QR (Common)
PR = SQ (Given)
∴ ΔPQR ≅ ΔSRQ (By SSS Congruence rule)
⇒ ∠PQR = ∠SRQ
Since adjacent angles of a parallelogram are supplementary. (Consecutive interior angles)
∠PQR + ∠SRQ= 180º
⇒ ∠PQR + ∠PQR= 180º
⇒ 2∠PQR= 180º
⇒ ∠PQR = 90º
Since PQRS is a parallelogram and one of its interior angles is 90º, PQRS is a rectangle.
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