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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 = 70o
Answer: 70o

Directions: Solve the following
Q 1: If the sum of the angles at the circumference and the centre is 120 degrees. Find the angle at centre.

Q 2: Consider triangle ABC in a circle, if one angle at the centre is 200 degrees, what is the other angle at the centre?

Q 3: If one angle at the circumference of a circle is 55 degrees , find the angle at centre.

Q 4:

Q 5:

Q 6: Find angle ACB.


Q 7:

Q 8:

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Question 10: This question is available to subscribers only!

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