Q) In the given figure, ABCD is a quadrilateral. Diagonal BD bisects ∠ B and ∠ D both. Prove that:
(i) Δ ABD ~ Δ CBD
(ii) AB = BC
Ans:
VIDEO SOLUTION
STEP BY STEP SOLUTION
Given: (i) ABCD is a quadrilateral and (ii) BD is angle bisector of ∠ B and ∠ D.
(i) To prove Δ ABD ~ Δ CBD:
Let’s compare Δ ABD with Δ CBD:
∠ ABD = ∠ CBD (given that BD is angle bisector of ∠ B)
∠ BDA = ∠ BDC (given that BD is angle bisector of ∠ D)
∴ by AA similarity criterion:
Δ ABD ~ Δ CBD
Hence Proved !
(ii) To prove AB = BC:
Let’s compare Δ ABD with Δ CBD:
∠ ABD = ∠ CBD (given that BD is angle bisector of ∠ B)
∠ BDA = ∠ BDC (given that BD is angle bisector of ∠ D)
BD = BD
∴ by ASA congruency criterion: Δ ABD 
Δ CBD
Next, by applying corresponding parts of congruent triangle or CPCT rule, we get:
AB = BC
Hence Proved !
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