Question

Prove that if the diagonals of a quadrilateral bisect each other then it is a parallelogram.

Answer 1

REF.Image.

let the quadrilateral be ABCD

AC and BD bisect each other at 0.

So, Ao = CO & BO = DO.

To be proved : ABCD is parallelogram.

consider $△DOC$ & $△BOA,$

$DO=BO$

$CO=AO$

also $\angle DOC=\angle BOA$

So, by SAS congruency, $△DOC\cong △BOA$

$\Rightarrow \angle CDO=\angle ABO$

These are pair of alternate angles.

So,$CD\parallel AB...\left(1\right)$

Similarly, $△AOD\cong △BOC$

which gives $AD\parallel BC...\left(2\right)$

$\therefore$ ABCD is a parallelogram