Answer :

Let us first draw a diagram for the above question.


In the above figure,


PQ and SR are two straight lines.


O is the common point.


AO is the angle bisector of POS.


NO is the angle bisector of SOQ


BO is the angle bisector of QOR


MO is the angle bisector of POR.


POR = SOQ [vertically opposite angles are always equal]


Similarly, POS = QOR [vertically opposite angles are always equal]


Let POR = SOQ = x and Let POS = QOR = y.


OR is perpendicular to line PQ.


So POR + QOR = 180° [as they form linear pair that is angles in a straight line always sum up to 180°]


MOR = MOP = 1/2 POR [as MO is the angle bisector]


Similarly, BOR = QOB = 1/2 QOR [as BO is the angle bisector]


MOB = MOR + BOR


= 1/2 POR + 1/2 QOR


= 1/2 × (POR + QOR)


= 1/2 × 180


MOB = 90°


So SOQ + QOR = 180° [as they form linear pair that is angles in a straight line always sum up to 180°]


NOS = NOQ = 1/2 SOQ [as NO is the angle bisector]


Similarly, QOB = BOR = 1/2 QOR [as BO is the angle bisector]


NOB = NOQ + QOB


= 1/2 SOQ + 1/2 QOR


= 1/2 × (SOQ + QOR)


= 1/2 × 180


NOB = 90°


Since the angle is 90°, it is proved that the bisectors of these angles are two perpendicular straight lines.


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