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

${\displaystyle \therefore \angle \text{BDC}=\frac{1}{2}\angle \text{BOC}=\frac{1}{2}\left(\text{x}\right)=\frac{1}{2}\text{x}}$

∠BDC = ∠ BEC = ∠ ${\displaystyle \frac{\text{x}}{2}}$ (angles in the same segment)

∠ ADB = AEP = 180 - ∠${\displaystyle \frac{\text{x}}{2}}$

Also , ∠BPC = ∠DPE = ∠ Y (Vertically opposite)

In quadrilateral ADPE ,

∠ADP + ∠DEP + ∠PEA + ∠EAD = 360°

180 - ∠${\displaystyle \frac{\text{x}}{2}}$ + ∠ Y + 180 - ∠ ${\displaystyle \frac{\text{x}}{2}}$ + z = 360°

- ∠ x + ∠ y + ∠ z = 0 

∠ x = ∠ y+ ∠ z