Question

In given figure if the diagonal BD of a quadrilateral ABCD bisects both $\angle B$ and $\angle D$, show that $\frac{AB}{BC}=\frac{AD}{CD}$.

Answer 1

Given A quadrilateral ABCD in which the diagonal BD bisects $\angle B$ and $\angle D$.

To prove $\frac{AB}{BC}=\frac{AD}{CD}$.

Construction Join AC intersecting BD in O.

Proof In $△ABC$, BO is the bisector of $\angle B$.

$\therefore$ $\frac{AO}{OC}=\frac{BA}{BC}$

$\Rightarrow$ $\frac{AO}{OC}=\frac{AB}{BC}$...........(i)

In $△ADC$, DO is the bisector of $\angle D$.

$\therefore$ $\frac{AO}{OC}=\frac{DA}{DC}$

$\Rightarrow$ $\frac{AO}{OC}=\frac{AD}{CD}$...........(ii)

From (i) and (ii), we get

$\frac{AB}{BC}=\frac{AD}{CD}$ $\left[Henceproved\right]$