High School Mathematics - 2 8.5 Angle at Centre Theorem
: Angle subtended by an arc at the centre is twice the angle subtended by the angle at the circumference.
Given: ACB is the major arc of the circle. Angles AOB and ACB stand on the same arc.
To prove: Angle AOB = 2 Angle ACB
Proof: From triangle AOC, x = a+a (Exterior angle of a triangle)
Hence x = 2a
From triangle OBC, y = b+b
Hence y = 2b
Angle AOB = x + y
Hence angAOB = 2(a+b)
Hence Angle AOB = 2 Angle ACB
Example: Find angle a.
Angle at circumference = 1/2 angle at centre.
Angle a = 70^{o} Answer: 70^{o}