Question

P is any point on the bisector of angle A OB. if PM BISECTS OA AND PN BISECTS OB. PROVE THAT PM = PN.

Answer 1

As the lines PL and PM are perpendicular to OA and AB respectively the angles PLO and PMO are both 90° and form a linear pair. POM and POL should form a triangle each. Also, PL=PM and they both have a common side(PO). This means that both the triangles are congruent by SAS(side,angle,side) which means that the remaining sides and angles must also be equal. So, angles AOP and BOP are equal which proves that OP is the bisector of angle AOB.