Prove that the if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.

Explanation

Take quadrilateral `ABCD` ,
`AC and BD` are diagonals which intersect at `O.`

In `triangle AOB and triangle AOD`

`DO=OB` (O is the midpoint)

`AO=AO` (Common side)

`angle AOB=angle AOD`( Right angle)

So, `triangle AOB~=triangle AOD`

So, `AB=AD`

Similarly, `AB=BC=CD=AD `can be proved which means that `ABCD` is a rhombus.