Question

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

Answer 1

Since $ABCD$ is a parallelogram, therefore, opposite sides will be equal.

$\Rightarrow AB=DC$ and $AD=BC$

Now, consider, $\Delta ADB$ and $\Delta BCA$

$AB=AB$ (Common)

$AD=BC$ (Opposite sides of parallelogram$

$BD=AC$ (given)

$\therefore \Delta ADB\cong \Delta BCD$ by SSS congruency rule.

$\therefore \angle DAB=\angle CBA$ (by CPCT)

But they also form a pair of supplementary angles. (Adjacent angles of a parallelogram are supplementary)

$\Rightarrow 2\angle DAB={180}^{\circ }$

$\Rightarrow \angle DAB=\angle CBA={90}^{\circ }$

Since, one angle of parallelogram $ABCD$ is ${90}^{\circ }$, therefore, it is a rectangle.