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Question
In following fig., O is the centre of the circle, prove that ∠x =∠ y + ∠ z.
Solution
Since arc BC makes ∠ BOC at the centre and ∠ BDC on the remaining part of the circle
`therefore angle "BDC" = 1/2 angle "BOC" = 1/2 ("x") = 1/2 "x"`
∠BDC = ∠ BEC = ∠ `"x"/2` (angles in the same segment)
∠ ADB = AEP = 180 - ∠`"x"/2`
Also , ∠BPC = ∠DPE = ∠ Y (Vertically opposite)
In quadrilateral ADPE ,
∠ADP + ∠DEP + ∠PEA + ∠EAD = 360°
180 - ∠`"x"/2` + ∠ Y + 180 - ∠ `"x"/2` + z = 360°
- ∠ x + ∠ y + ∠ z = 0
∠ x = ∠ y+ ∠ z
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