Question

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

Please don't post irrevelent answer...​

Answer 1

Destination

Given AB and CD are straight lines which intersect at O. OP is the bisector of ∠AO.

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 rate opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle. Similarly, the bisection of ∠AOD also bisects the ∠BOC.

Answer 2

Given AB and CD are straight lines which intersect at O. OP is the bisector of ∠AO.

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 rate opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle. Similarly, the bisection of ∠AOD also bisects the ∠BOC.

i am answering this via phone so i can't upload the picture

but i can dictate

pq is a horizontal line(p on left side)

ab and cd forms a cross over it

with a being on the left side of the page and d on the right