Revision notes
Learn the circle theorems: the angle at the centre, angles in the same segment and the angle in a semicircle.
Bola and Musa sit at different seats on a stand shaped like part of a circle that passes through both goalposts. Surprisingly, they see the goalmouth at exactly the same angle. A circle theorem explains why.
Take two points A and B on a circle with centre O. The arc AB subtends (makes) an angle ∠AOB at the centre and an angle ∠ACB at any point C on the rest of the circle.
The angle at the centre is twice the angle at the circumference: ∠AOB = 2 × ∠ACB.
So if ∠ACB = 35°, then ∠AOB = 70°. If ∠AOB = 140°, then ∠ACB = 70°.
🔑 Key idea: Angle at centre = 2 × angle at circumference (standing on the same arc).
Angles in the same segment are equal. Every angle standing on arc AB from the same side is half of ∠AOB, so they are all the same — just like the fans in the stand.
The angle in a semicircle is 90°. If AB is a diameter, ∠AOB is a straight angle, 180°. Half of 180° is 90°.
So in triangle ABC with diameter AB, if ∠CAB = 30° then ∠ABC = 180° − 90° − 30° = 60°.
🔑 Key idea: Same segment → equal angles. Angle in a semicircle = 90°.
All radii are equal, so triangle OAB (two radii and a chord) is isosceles and its base angles are equal.
If ∠AOB = 100°, then ∠OAB = ∠OBA = (180° − 100°) ÷ 2 = 40°.
🔑 Key idea: Two radii → isosceles triangle → equal base angles.
O is the centre of a circle. A, B and C are points on the circle, with C on the major arc. ∠ACB = 60°. Find ∠AOB.
Step 1: Same arc AB
∠AOB (at the centre) and ∠ACB (at the circumference) both stand on arc AB.
Step 2: Double it
∠AOB = 2 × 60° = 120°
Answer: 120°
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