In the given figure, ABCD is a quadrilateral. Diagonal BD bisects $\angle B$ and $\angle D$ both. Prove that :
(i) $\triangle ABD \sim \triangle CBD$
(ii) $AB = BC$
Generated by claude-sonnet-4-6 · 2026-06-15 10:28 · grounding rag
Model Answer
In △ABD and △CBD:
∠ABD = ∠CBD (BD bisects ∠B, given)
∠ADB = ∠CDB (BD bisects ∠D, given)
BD = BD (common side)
(i) By AA similarity criterion,
△ABD ~ △CBD
(ii) Since △ABD ~ △CBD, their corresponding sides are proportional:
$$\frac{AB}{CB} = \frac{BD}{BD} = \frac{AD}{CD}$$
Since BD/BD = 1, we get AB/CB = 1, i.e., AB = BC.
Source: Chapter 6, Section 6.4 (AA Similarity Criterion)
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Explanation
- For part (i), two pairs of equal angles (AA) are sufficient to prove similarity — you don't need all three angles explicitly.
- For part (ii), once similarity is established, corresponding sides are proportional. The ratio BD/BD = 1 forces all ratios to equal 1, giving AB = BC directly. Examiners expect you to link part (ii) back to the similarity result from part (i).
- Write the similarity statement with correct vertex correspondence: △ABD ~ △CBD (A↔C, B↔B, D↔D).