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

Given,

Lines A0B and COD intersect at point O, such that

∠AOC - ∠BOO

Also OE is the bisector of ADC and OF is the bisector of BOD

To prove: EOF is a straight line, vertically opposite angles are equal

AOD = BOC = 5x  ... (1)

Also,

AOC + BOD

2 AOD = 2 DOF ... (2)

We know,

Sum of the angles around a point is 360

2AOD + 2AOE + 2DOF = 360

AOD + AOE + DOF = 180

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

Given that: - AB and CD intersect each other at O

OE bisects COB

To prove: AOF = DOF

Proof: OE bisects COB

COE = EOB = x

Vertically opposite angles are equal

BOE = AOF = x  ... (1)

COE = DOF = x .... (2)

From (1) and (2),

∠AOF = ∠DOF = x

Hence Proved.