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If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

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Given: AB and CD are straight lines which intersect at O. 

OP is the bisector of ∠AOC. 

To Prove : OQ is the bisector of ∠BOD 

Proof :

AB, CD and PQ are straight lines which intersect in O. 

Vertically opposite angles: ∠AOP = ∠BOQ 

Vertically opposite angles: ∠COP = ∠DOQ 

OP is the bisector of ∠AOC  : ∠AOP = ∠COP 

Therefore,  ∠BOQ  = ∠DOQ 

Hence, OQ is the bisector of ∠BOD.

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