Question

If $ABCD$ is a quadrilateral such that diagonal $AC$ bisects the angles $A$ and $C$,then prove that $AB=AD$ and $CB=CD$.

Answer 1

Given:

In quadrilateral $ABCD$ diagonal $AC$ bisects the angles $A$ and $C$.

To prove: $AB=AD$ and $CB=CD$

Proof:

In$\Delta ADC\text{and}\Delta ABC,$
$\angle DAC=\angle BAC$ [$\because AC$ is the bisector of $\angle A$ and $\angle C$]
$\angle DCA=\angle BCA$ [ $\because AC$ is the bisector of $\angle A$ and $\angle C]$
$AC=AC$ [common side]
$\Delta ADC\cong \Delta ABC$ [by ASA congruence rule]

By C.P.C.T,
$AD=AB$ ---(1)
$CD=CB$ ---(2)

Hence, proved.