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Prove-that-bisectors-pair-vertically-opposite-angles-are-same-straight-line - Mathematics

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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

Concept: Concept of Parallel Lines
  Is there an error in this question or solution?

APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 10 Lines and Angles
Exercise 10.3 | Q 12 | Page 23

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