In the diagram below, mnpq is a parallelogram whose diagonals are perpendicular. prove: mnpq is a rhombus.

Given : MNPQ is a parallelogram whose diagonals are perpendicular.

To prove :  MNPQ is a rhombus.

Proof:

In parallelogram MNPQ,

R is the intersection point of the diagonals MP and NQ( shown in below diagram)

(Because, the diagonals of parallelogram bisects each other)

(Right angles )

 (Reflexive)

Thus, By SAS postulate of congruence,

By CPCTC,

Similarly,

We can prove,

By CPCTC,

But, By the definition of parallelogram,

and

⇒

All four side of parallelogram MNQP are congruent.

⇒ Parallelogram MNPQ is a rhombus.

Hence, proved.