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In the figure, P is the centre of the ci...

In the figure, P is the centre of the circel. Prove that : `angle XPZ=2(angleXZY+angleYXZ)`.

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Since, arc XY subtends `angleXPY` at the centre and `angle XZY` at a point Z in the remaining part of the circle.
`therefore angle XPY =2angle XZY` ........(1)
Similarly, arc YZ subtends `angleYPZ` at the centre and `angleYXZ` at a point X in the remaining part of the circle.
`therefore angle YPZ=2angleYXZ`
Adding eqs. (1) and (2) , we have
`angle XPY+angle YPZ=2angle XZY+2angle YXZ`
`rArr angle XPZ=2(angle XZY+angle YXZ)`
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