In a circle with centre O , XZ is any chord of the circle If Y is a point - Maths - Circles

Question

In a circle with centre O , XZ is any chord of the circle If Y is a
point on the minor arc then show that angle XYZ=2(angle XZY+angle
YXZ)

Manbar Singh · Accepted Answer

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Join OY
We know that,angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the circle
Consider the minor arc XY, that subtends ∠XOY at the centre and ∠XZY at the circle
Now,  ∠XOY = 2 ∠XZY    
1Consider the minor arc YZ, that subtends ∠YOZ at the centre and ∠YXZ at the circle
Now,  ∠YOZ = 2 ∠YXZ    
2adding 1 and 2, wqe get   ∠XOY + ∠YOZ  = 2∠XZY  +  2 ∠YXZ⇒∠XOZ = 2∠XZY + YXZ

In a circle with centre O , XZ is any chord of the circle. If Y is a point on the minor arc then show that angle XYZ=2(angle XZY+angle YXZ)