 # ABCD is a quadrilateral in which AB = BC `
Question:

ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.

Thinking Process

Firstly, use the property that if two sides of a triangle are equal, then their opposite angles are equal. Further, show that  ΔBAD and ΔBCD are

congruent by SAS rule.

Solution:

Given $A B C D$ is a quadrilateral in which $A B=B C$ and $A D=C D$.

To show $B D$ bisects both the angles $A B C$ and $A D C$.

Proof Since, $A B=B C$  (given)

$\therefore$ $\angle 2=\angle 1$ $\ldots$ (i)

[angles opposite to equal sides are equal] On adding Egs. (0) and (ii), we get

$\angle 2+\angle 4=\angle 1+\angle 3$

$\Rightarrow \quad \angle B C D=\angle B A D \quad \ldots$ (iii)

In $\triangle B A D$ and $\triangle B C D$,

$A B=B C$ [given]

$\angle B A D=\angle B C D$ [from Eq. (iii)]

and $A D=C D$[given]

$\therefore$ $\triangle B A D \cong \triangle B C D$ [by SAS congruence rule]

Hence, $\angle A B D=\angle C B D$ and $\angle A D B=\angle C D B$ i.e., $B D$ bisects the angles $A B C$ and $A D C$.