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

Explanation

GIVEN Two line AB and CD intersect at a point O. Also, OE and OF are th bisectors of `angleAOC` and `angleBOD` respectively.
TO PROVE EOF is a straight line.
PROOF Since the sum of all the angles around a point is `360^(@)`, we have:
`angleAOC+angleBOC+angleBOD+angleAOD=360^(@)`
`implies angleAOC+2angleBOC+angleBOD=360^(@)`
`[ :' angleAOD=angleBOC ("vert. opp." angles)]`
`implies 2angleEOC+2angleBOC+2angleBOF=360^(@)`
[ `:'` OE is the bisector of `angleAOC` and OF is the bisector of `angleBOD`]
`implies angleEOC+angleBOC+angleBOF=180^(@)`
`implies angleEOF=180^(@)`, i.e. `angleEOF` is a straight angle.
Hence, EOF is a straight line.