Question

If the diagonals of a quadrilateral bisect each other, then prove that it is a parallelogram.

Answer 1

$ABCD$ is an quadrilateral with $AC$ and $BD$ are diagonals intersecting at $O.$

It is given that diagonals bisect each other.

$\therefore$ $OA=OC$ and $OB=OD$

In $△AOD$ and $△COB$

$\Rightarrow$ $OA=OC$ [ Given ]

$\Rightarrow$ $\angle AOD=\angle COB$ [ Vertically opposite angles ]

$\Rightarrow$ $OD=OB$ [ Given ]

$\Rightarrow$ $△AOD\cong △COB$ [ By SAS Congruence rule ]

$\therefore$ $\angle OAD=\angle OCB$ [ CPCT ] ----- ( 1 )

Similarly, we can prove

$\Rightarrow$ $△AOB\cong △COD$

$\Rightarrow$ $\angle ABO=\angle CDO$ [ CPCT ] ---- ( 2 )

For lines $AB$ and $CD$ with transversal $BD$,

$\Rightarrow$ $\angle ABO$ and $\angle CDO$ are alternate angles and are equal.

$\therefore$ Lines are parallel i.e. $AB\parallel CD$

For lines $AD$ and $BC$, with transversal $AC,$

$\Rightarrow$ $\angle OAD$ and $△OCB$ are alternate angles and are equal.

$\therefore$ Lines are parallel i.e. $AD\parallel BC$

Thus, in $ABCD,$ both pairs of opposite sides are parallel.

$\therefore$ $ABCD$ is a parallelogram.