Question

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

Answer 1

By considering $△ABC$ and $△ADC$

We know that $AC$ bisects at $\angle A$

So we get

$\angle BAC=\angle DAC$

We know that $AC$ is common i.e. $AC=AC$

From the figure, we know that $AC$ bisects at $\angle C$

$\angle BCA=\angle DCA$

By $ASA$ congruence criterion we get

$△ABC\cong △ADC$

Therefore, it is proved that $AB=AD$ and $CB=CD(c.p.c.t)$.