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:

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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