Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on

the remaining part of the circle.

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Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on

the remaining part of the circle.

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Now join A to O and B to O.Angle AOB is a angle at centre .Let any point C on any part of the circle.Angle ACB will be any angle at the remaining part of the circle.

Now we have to prove:angle AOB=2angle ACB

We will draw aline passing through the centre from angle ACB.It will touch AB at point D.

angle ACO+angle OAC=angle AOD

Since AO=CO(radius)

Therefore, angleACO=angleOAC

2angle ACO=angleAOD........................(i)

angle BCO+angle OBC=angle BOD

Since BO=CO(radius)

Therefore, angleBCO=angleOBC

2angle BCO=angle BOD........................(ii)

Adding (i)and(ii)

2angleBCO+2angleACO=angle BOD+angle AOD

Therefore,angle AOB=2angleACB(BCO+ACO=ACB)