Q. 64.2( 56 Votes )

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Answer :

To Prove: If diagonals of a parallelogram are equal, the quadrilateral is a rectangle.


Given: Diagonals area equal


Concept Used:


Rectangle: Opposite sides are equal, and all angles are right angles.


SSS Theorem: If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.


Opposite angles of a parallelogram are supplementary.


Diagram:



Explanation:


In ΔABD and Δ BCD,


AB = CD [Opposite sides of a parallelogram are equal]


AD = BC [Opposite sides of a parallelogram are equal]


BD = Common


Therefore, ΔABD and ΔBCD are congruent.


As the triangles are congruent,


BAD = BCD [By C.P.C.T]


Now, we also know that,


Opposite angles of a parallelogram are supplementary.


Therefore,


BAD + BCD = 180˚


2 BAD = 180˚


BAD = 90˚ = BCD


And similarly


ADC = ABC = 90˚


Now, all angles of the parallelogram are right angles and opposite sides are equal.


The parallelogram is a rectangle.


Hence, Proved.


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