Angle Subtended by the Arc to the Centre

In the above figure, ∠ABC is inscribed in arc ABC.

If ∠ABC = 60°. find m ∠AOC.

Solution:

∠ABC = `1/2` m(arc AXC)   ......`square`

60° = `1/2` m(arc AXC) 

`square` = m(arc AXC) 

But m ∠AOC = \[\boxed{m(arc ....)}\]   ......(Property of central angle)

∴ m ∠AOC = `square`

In the given figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that,

1. `square`WZPT is cyclic.
2. Points X, Z, T, Y are concyclic.

In the following figure, O is the centre of the circle. ∠ABC is inscribed in arc ABC and  ∠ ABC = 65°. Complete the following activity to find the measure of ∠AOC. 

∠ABC = `1/2`m ______  (Inscribed angle theorem) 
______ × 2 = m(arc AXC)  
m(arc AXC) = _______
∠AOC = m(arc AXC)  (Definition of measure of an arc)  
∠AOC = ______