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

Given that,

Lines AOB and COD intersect at point O such that,

∠AOC = ∠BOD

Also,

OE is the bisector of ∠AOC and OF is the bisector of ∠BOD

To prove: EOF is a straight line.

∠AOD = ∠BOC = 5x (Vertically opposite angle) (i)

∠AOC + ∠BOD

2∠AOE = 2∠DOF (ii)

Sum of the angles around a point is 360^o

2∠AOD + 2∠AOE + 2∠DOF = 360^o

∠AOD + ∠AOF + ∠DOF = 180^o

From this we can conclude that EOF is a straight line.