Example 1 · level 1
Arc AB subtends ∠AOB = 144° at the centre O. Find ∠ACB, where C is a point on the remaining part of the circle (in degrees).
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- ∠ACB = ½ × 144° = 72°.
Answer: 72
CLASS 9 MATHS · CIRCLES
Angles in a circle and cyclic quadrilaterals
The angle at the centre is twice the angle at the circumference. An angle in a semicircle is 90°. Opposite angles of a cyclic quadrilateral add to 180°.
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Worked examples
Example 2 · level 2
AB is a diameter of a circle and C is a point on the circle. If ∠CAB = 24°, find ∠CBA (in degrees).
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- ∠ACB = 90° (angle in a semicircle).
- ∠CBA = 180° − 90° − 24° = 66°.
Answer: 66
Example 3 · level 3
ABCD is a cyclic quadrilateral with ∠A = 105° and ∠B = 51°. Find ∠C (in degrees).
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- ∠C + ∠A = 180°.
- ∠C = 180° − 105° = 75°.
Answer: 75