Question

$AP$ and $BQ$ are the bisectors of the two alternate interior angles formed by the intersection of transversal $t$ with parallel lines $l$ and $m$. Show that $AP\parallel BQ$

Answer 1

$\because l||m$ and $t$ is the transversal

$\angle MAB=\angle SBA$ [Alt. $\angle s$]

$\frac{1}{2}\angle MAB=\frac{1}{2}\angle SBA\Rightarrow \angle PAB=\angle QBA$

But, $\angle PAB$ and $\angle QBA$ are alternate angles.

Hence, $AP||BQ$