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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.



Let the centre be O and the segment be AB.
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)

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