Parallelograms Converse of Theorem 5

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Statement

If the diagonals of a parallelogram are perpendicular then it is a rhombus.

Given:

ABCD is a parallelogram in which AC and BD are perpendicular to each other.

To prove:

ABCD is a rhombus.

Proof:

Let AC and BD intersect at right angles at O.

In triangles AOD and COD,

AO=OC (diagonals bisect each other)

OD=OD (common side)

AD = DC

i.e., the adjacent sides are equal.

By definition, ABCD is a rhombus.

Hence the theorem is proved.