Question

In quadrilateral ABCD, AB = BC and AD = CD. Show that BD bisects ∠ABC and ∠ADC both.

Answer 1

Given: AB = BC and AD = CD in quadrilateral ABCD.

To Prove: BD bisects ∠ABC and BD bisects ∠ADC (i.e., ∠ABD = ∠DBC and ∠ADB = ∠BDC).

Proof (Step-wise):

1. Consider triangles △ABD and △CBD.

2. AB = BC.   ...(Given)

3. AD = CD.   ...(Given)

4. BD = BD.   ...(Common side)

5. From (2), (3) and (4), the three sides of △ABD are respectively equal to the three sides of △CBD. Hence △ABD ≅ △CBD by SSS congruence.

6. By corresponding parts of congruent triangles, ∠ABD = ∠DBC and ∠ADB = ∠BDC. This shows BD bisects ∠ABC and BD bisects ∠ADC. (Compare this use of triangle congruence and CPCTC with the congruence/angle-equality reasoning in the provided geometry notes.)

Therefore, BD bisects both ∠ABC and ∠ADC.