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

Explanation

`angle PBQ=anglePCQ` (angles of same segment)
Arc PQ subtends angles at
centre `=anglePOQ`
remaining circle `=anglePBQ` or `angle PCQ`
`therefore angle PBQ=anglePCQ=(1)/(2)anglePOQ=(z)/(2)`
Now, `angle BDC=angle PDQ=y`
`angle ABD=180^@-anglePBQ=180^@-(z)/(2)`
`angle ACD =180^@-angle PCQ=180^@-(z)/(2)`
In square `ABDC`,
`angle ABD+angle BDC+angle DCA+angle CAB=360^@`
`rArr(180^@-(z)/(2))+y+(180^@-(z)/(2))+x=360^@`
`rArr-z+y+x=0`
`rArrx+y=z`