Question

If the diagonals of a parallelogram are equal in length, then prove that the parallelogram is a rectangle.

Answer 1

Given $ABCD$ is parallelogram.

$AC$ and $BD$ are diagonals of parallelogram such that $AC=BD$

In $△ABD\&△ACD$

$AC=BD$

$AB=DC$ (opposite sides of parallelogram)

$AD=AD$ (common side)

$\therefore$$\Delta ABD$ $\cong$ $\Delta DCA$ (by $SSS$ criterion )

$\angle BAD$ = $\angle CDA$ (Corresponding parts of congruent triangles are equal)

$\angle BAD+\angle CDA={180}^{\circ }$(adjacent angles)

$⟹\angle BAD=\angle CDA={90}^o$ ( as they are congruent and supplementary.)

Parallelogram $ABCD$ is a rectangle since any parallelogram with one right interior angle is rectangle.