Question

$\text{ABCD is a quadrilateral such that diagonal AC bisects that angles}\angle Aand\angle C.\text{Prove that AB = AD and CB = CD.}$

Answer 1

Given: In quadrilateral ABCD, AC bisects the angles ∠ A and ∠ C.

To prove: AB = AD and CB = CD

Proof:

In ∆ ABC and ∆ ADC,

∠ BAC = ∠ DAC (Given, AC bisects the angles ∠ A)

AC = AC (Common side)

∠ BCA = ∠ DCA (Given, AC bisects the angles ∠ C)

∴ By ASA congruence criteria,

∆ ABC ≅∆ ADC

Hence, AB = AD and CB = CD . (CPCT)