Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Explanation

Let AB and CD are straight lines intersecting at O. OX the bisector of angles ∠AOC and OY are the OY is the bisector of ∠BOD
Understand that, OY is the bisector of ∠BOD.
`/_1=/_6`..(1)
Understand that, OX is the bisector of `/_AOC
/_3=/_4`......(2)
Observe that, vertically opposite angles:
` /_2 =/_5` .......(3)
Understand that, the sum of the angles formed at a point is `360@`.
`/_(1+2+3+4+5+6) =180@`
Use equation (1), equation (2) and equation (3)
`/_ (1+2+3+3+2+1) =360^@`
`=2 /_(1+2+3)=360^@`.
Hence, `∠DOY +∠AOD+ ∠AOX=180^@`
⇒`XOY =180 ^@`
Therefore, the bisectors of pair of vertically opposite angles are on the same straight line.