Question

if 2 straight lines intersect each other then prove that ray opposite two the bisector of the angles so form bisects the vertically opposite angle.

Answer 1

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.
∠ AOP = ∠ BOQ (vertically opposite angles)
∠ COP = ∠ DOQ (vertically opposite angles)
∠ AOP = ∠ COP (OP is the bisector of ∠ AOC)
∴ ∠ BOQ = ∠ DOQ
Thus, the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.
Similarly, the bisector of ∠ AOD also bisects the ∠ BOC.