Question

Prove that if two parallel lines are intersected by a transversal, then the bisectors of the interior angles on the same side of the transversal intersect at right angles.

Answer 1

Let the angle at which the transversal intersects the lines be $\theta$

\So, $\angle BAD=\theta ,\angle EBA=180-\theta$

$OB$ bisects the $\angle EBA$ and $OA$ bisects the $\angle BAD$.

consider $△AOB,\angle BAO=\frac{\theta }{2}$

$\angle OBA=\frac{1}{2}(180-\theta )=90-\frac{\theta }{2}$

$\angle BAO+\angle OBA+\angle BOA=180°$

$⟹90-\frac{\theta }{2}+\frac{\theta }{2}+\angle AOB=180°$

$⟹\angle AOB=90°$

$\therefore$ the bisectors of internal angles on the same side of the transversal intersects at right angles.