Question

Prove that if a diagonal of a quadrilateral bisect each other, it is a parallelogram.

Answer 1

Let $ABCD$ be a quadrilateral such that $BO=OD$ and $AO=OC$

Consider $\Delta AOD$ and $\Delta BOC$

$BO=OD$ [Given]

$AO=OC$ [Given]

$\angle DOA=\angle BOC$ [Vertically Opposite Angles]

Thus, $\Delta AOD\cong \Delta COB$ by SAS rule.

$\Rightarrow AD=BC$ [By CPCT] ...(i)

and

$\Rightarrow \angle ADO=\angle CBO$

But they also form a pair of equal alternate interior angles.

$\therefore AD\parallel BC$ ...(ii)

From (i) and (ii), one pair of opposite sides are equal and parallel and hence,

$ABCD$ is a parallelogram.