Parallelograms Converse of Theorem 6

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Statement

If in a parallelogram, the diagonals are equal and perpendicular, then it is a square.

Given:

ABCD is a parallelogram.

AC=BD and AC ^ BD

To prove:

ABCD is a square.

Proof:

Since the diagonals AC and BD are equal,

ABCD is a rectangle - - -(i)

(Diagonal property of rectangle)

Since the diagonals are perpendicular to each other.

ABCD is a rhombus.

AB=AD - - -(ii)

ABCD is a rectangle. (from i)

With consecutive sides equal. (from ii)

ABCD is a square. (by definition of a square)

Hence the theorem is proved.