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In figure O is the centre of the circle,...

In figure O is the centre of the circle, prove that `anglez=anglex+angley`.

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`angle1 =angle2 ` (angles in the same segment of a circle are equal
`anglez=2angle1` (subtended by an arc AB)
`therefore anglez=angle1+angle 2" "(because angle1=angle2)`
`rArranglez=angle y-angle3+angle2" "(because "exterior of" angley=angle1+angle3)`
`=angley-angle3+anglex+angle3" "(because "exteriorof" angle2=anglex+angle3)`

Hence, `anglez=anglex+angley`.
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