Question

if 2 ll lines are intersected by a transversal prove that the bisector of the interior angles on the same side of transversal intersect each other at right angles

Answer 1

if a transversal intersects two parallel lines then the sum of angles on the same side are equal to 180 degrees. known to us 
if they are bisected that is made into half then the sum of angles will be equal to 90 degrees .
hope this helps u

Answer 2

Solutions:

We know that the sum of interior angles on the same side of the transversal is 180°.

Hence, ∠BMN + ∠DNM = 180°

=> 1/2∠BMN + 1/2∠DNM = 90°

=> ∠PMN + ∠PNM = 90°

=> ∠1 + ∠2 = 90° ............. (i)

In △PMN, we have

∠1 + ∠2 + ∠3 = 180° ......... (ii)

From (i) and (ii), we have

90° + ∠3 = 180°

=> ∠3 = 90°

=> PM and PN intersect at right angles.