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

#### Solution

Given,

Lines *A**O**B *and *C**O**D *intersect at point O such that

`∠`*A**O**C *= `∠`*B**O**D*

Also OE is the bisector `∠`*A**D**C *and OF is the bisector `∠`*B**O**D*

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

*`∠`AOD *= `∠`*BOC *= 5*x *.......(1)

Also `∠`*A**O**C *+ `∠`*B**O**D*

⇒ 2`∠`*AOE *= 2`∠`*DOF *.......(2)

Sum of the angles around a point is 360°

⇒ 2`∠`*AOD *+ 2`∠`*AOE *+ 2`∠`*DOF *= 360°

⇒`∠`*AOD *+ `∠`*AOF *+ `∠`*DOF *= 180°

From this we conclude that *E**O**F *is a straight line.

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

OE bisects `∠`*C**O**B*

To prove: `∠`*A**O**F *= `∠`*DO**F*

Proof: *O**E *bisects `∠`*C**O**B*

*`∠`C**O**E *= `∠`*E**O**B *= *x*

Vertically opposite angles are equal

*`∠`BOE *= `∠`*AOF *= *x *.......(1)

*`∠`COE *= `∠`*DOF *= *x *.......(2)

From (1) and (2)

*`∠`A**O**F *= `∠`*DO**F *= *x*