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Question

Let AB and CD be straight lines intersecting at O. Let OP be the bisector of BOD and OQ be the bisector of AOC. Prove that Q,O,P are collinear.
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Solution

AB and CD be straight lines intersecting at O.
AOC=BOD ....... [Opposite angles] ........ (i)
OP bisects BOD ........ [Given]
BOP=POD
BOP=12BOD ......... (ii)
OQ bisects AOC ....... [Given]
COQ=12AOC ......... (iii)
Now, BOP+BOC+COQ
=12BOD+BOC+12AOC
=12(AOC+BOD)+BOC
=AOC+BOC ........ From (i)
=180o ..... [Since AB is a straight line and O is a point on AB]
BOP+BOC+COQ=180o
BOP,BOC,COQ form a linear pair.
Thus, POQ=180o
Hence, Q,O,P are collinear

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