Question

In quadrilateral ABCD, the diagonal AC bisects ∠A and ∠C both. Prove that AB = AD and CB = CD.

Answer 1

Given: AC bisects ∠A and ∠C in quadrilateral ABCD (so ∠BAC = ∠CAD and ∠BCA = ∠DCA)

To Prove: AB = AD and CB = CD

Proof (Step-wise):

1. Consider triangles △ABC and △ADC.

2. From the hypothesis AC bisects ∠A, we have ∠BAC = ∠CAD.

3. From the hypothesis AC bisects ∠C, we have ∠BCA = ∠DCA.

4. AC is common to both triangles, so AC = AC.

5. Thus, in △ABC and △ADC we have:

∠BAC = ∠CAD

AC = AC

And ∠BCA = ∠DCA

Therefore, the two triangles are congruent by ASA congruence.

6. By CPCT (corresponding parts of congruent triangles), AB = AD and BC = DC.

AB = AD and CB = CD, as required.