If two parallel lines are intersected by a transversal, prove that the bisectors of the interior angles on the same side of transversal intersect each other at right angles.

Explanation

Let the angle at which the transversal intersects the lines be θ
So, ∠BAD=θ,∠EBA='180@'−θ
OB bisects the ∠EBA and OA bisects the ∠BAD.
consider △AOB,'∠BAO= θ/2'
'∠OBA= 1/2 (180−θ)=90− θ/2'
'∠BAO+∠OBA+∠BOA=180@'
⟹'90−θ/2 +θ/2 +∠AOB=180@'
⟹'∠AOB=90@'
∴ the bisectors of internal angles on the same side of the transversal intersects at right angles.