Question

if two straight lines intersect each other then prove that the ray opposite to the bisector of one of the angles so formed bisect 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.