Question

In the figure, bisectors $AP$ and $BQ$ of the alternate interior angles are parallel. Show that $l//m$.

Answer 1

$A$ is the bisector of $\angle MAB$ and $BQ$ is the bisector of

$\angle SBA$. We are given that $AP||BQ$

As $AP||BQ$, So $\angle 2=\angle 3$[Alt. $\angle s]$

$\therefore 2\angle 2=2\angle 3$

$\angle 2+\angle 2=\angle 3+\angle 3$

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

and $\angle 3=\angle 4]$

$\Rightarrow \angle MAB=\angle SBA$

But, these are alternate angles. Hence, the lines $l$ and $m$ are parallel, i.e., $l||m$