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Circle Theorems Worksheet
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Circle Theorems Worksheet
📚 Part 1: Circle Theorem Knowledge
1. The angle at the centre is twice the angle at the circumference standing on the same arc. If the angle at the centre is 120°, what is the angle at the circumference?
30°
60°
90°
240°
2. In a cyclic quadrilateral, opposite angles sum to:
90°
180°
270°
360°
3. Which of these statements about tangents are true? (Select all that apply)
A tangent is perpendicular to the radius at the point of contact
Two tangents from an external point are equal in length
A tangent passes through the centre of the circle
The angle between a tangent and a chord equals the angle in the alternate segment
4. Angles in the same segment are:
Always 90°
Equal to each other
Sum to 180°
Always acute angles
✏️ Part 2: Problem Solving
5. In a circle with centre O, the angle AOB at the centre is 84°. Calculate the angle ACB at the circumference standing on the same arc AB.
Answer: ∠ACB = _______
6. In cyclic quadrilateral PQRS, angle P = 75° and angle Q = 110°. Find angles R and S.
∠R = _______ and ∠S = _______
7. Two angles in the same segment of a circle are (3x + 20)° and (5x - 8)°. Find the value of x.
x = _______
8. A tangent to a circle makes an angle of 35° with a chord at the point of contact. What is the angle in the alternate segment?
Angle in alternate segment = _______
🎯 Part 3: Reasoning and Proof
9. Draw a diagram to show why angles in the same segment are equal. Label your diagram clearly and explain your reasoning.
10. Explain why the angle between a tangent and radius at the point of contact is always 90°. Use correct mathematical vocabulary in your answer.
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