Question

In the given figure, ABCD is a quadrilateral with $\angle 1=\angle 2$ and $\angle 3=\angle ABC$ and $\angle 4=\angle BAD$. Also, AP and BQ are produced to points E and F, respectively. Then, we can say that

Answer 1

Given, ABCD is a quadrilateral with $\angle 1=\angle 2,\angle 3=\angle ABC$ and $\angle 4=\angle BAD$
In $\Delta APB$ and $\Delta EPC$,
$\angle APB=\angle CPE$ [vertically opposite angles]
$\angle PCE=\angle 3=\angle PBA$ [given]
$\therefore \Delta APB\sim \Delta EPC$ [by AA similarity]
$\Rightarrow \angle PEC=\angle 1$ ...(i)
Similarly, in $\Delta FQD$ and $\Delta BQA$,
$\angle QFD=\angle 2$ ...(ii)
From Eqs. (i) and (ii), we get
$\angle PEC=\angle QFD[\because \angle 1=\angle 2,given]$
$\Rightarrow \angle OFE=\angle OEF$
$\Rightarrow OF=OE$
$\therefore \Delta OFE$ is an isosceles triangle.