Question

In the given figure, ABCD is a quadrilateral. Diagonal BD bisects ∠B and ∠D both.

Prove that: 

1. ΔABD ~ ΔCBD 
2. AB = BC

Answer 1

Given:

ABCD is a quadrilateral.

Diagonal BD bisects both ∠B and ∠D.

∠ABD = ∠DBC and ∠ADB = ∠CDB

i. To prove △ABD ∼ △CBD:

In △ABD and △CBD

∠ABD = ∠DBC (Given, BD bisects ∠B)

∠ADB = ∠CDB (Given, BD bisects ∠D)

BD = BD (Common side)

By ASA (Angle-Side-Angle) criterion,

△ABD ∼ △CBD

ii. To prove AB = BC:

From similarity of triangles △ABD and △CBD:

`(AB)/(CB) = (BD)/(BD) = 1`

⇒ AB = BC