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