Question

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

Answer 1

Given,

Lines  AOB and COD intersect at point O such that

`∠`AOC = `∠`BOD

Also OE is the bisector `∠`ADC and OF is the bisector `∠`BOD

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

`∠`AOD = `∠`BOC = 5x        .......(1)

Also `∠`AOC + `∠`BOD

⇒ 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 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