If P is a point in the interior of angle AOB. If PL is perpendicular to OA and PM is perpendicular to OB such that PL=PM, show that OP is the bisector of angle AOB.
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It is proved that OP is the bisector
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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.
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