Question

In following  fig., O is the centre of the circle, prove that ∠x =∠ y + ∠ z. 

Answer 1

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