Circle Theorems worksheet
10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.
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2 marks
\(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
(a)Calculate the size of the angle marked \(x\).
(b)Give a reason for your answer.
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2 marks
\(A\), \(B\), \(C\) and \(D\) are points on a circle.
(a)Write down the size of the angle marked \(x\).
(b)Explain your answer.
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4 marks
\(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
\(AB\) is a diameter of the circle.
(a)State the size of angle \(ACB\).
(b)Explain your answer.
(c)Calculate the size of the angle marked \(x\).
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4 marks
\(A\), \(B\), \(C\) and \(D\) are points on a circle.
(a)Work out the size of the angle marked \(x\).
(b)Give a reason for your answer.
(c)Work out the size of angle \(ADC\).
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4 marks
\(A\) and \(B\) are points on a circle, centre \(O\).
\(PA\) and \(PB\) are tangents to the circle.
(a)Write down the size of angle \(OAP\).
(b)Give a reason for your answer.
(c)Calculate the size of the angle marked \(x\).
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3 marks
\(PA\) and \(PB\) are tangents to a circle, touching it at \(A\) and at \(B\).
(a)Find the size of the angle marked \(x\).
(b)Give a reason why triangle \(ABP\) is isosceles.
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2 marks
\(T\), \(A\) and \(B\) are points on a circle.
The straight line through \(T\) is a tangent to the circle.
(a)Write down the size of the angle marked \(x\).
(b)Explain your answer.
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2 marks
\(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
(a)Calculate the size of the angle marked \(x\).
(b)Give a reason for your answer.
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2 marks
\(A\), \(B\), \(C\) and \(D\) are points on a circle.
(a)Write down the size of the angle marked \(x\).
(b)Explain your answer.
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4 marks
\(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
\(AB\) is a diameter of the circle.
(a)Write down the size of angle \(ACB\).
(b)Give a reason for your answer.
(c)Find the size of the angle marked \(x\).
Answers
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(a) \(65^\circ\)
(b) The angle at the centre is twice the angle at the circumference -
(a) \(35^\circ\)
(b) Angles in the same segment are equal -
(a) \(90^\circ\)
(b) The angle in a semicircle is 90 degrees
(c) \(31^\circ\) -
(a) \(102^\circ\)
(b) Opposite angles of a cyclic quadrilateral add up to 180 degrees
(c) \(79^\circ\) -
(a) \(90^\circ\)
(b) The angle between a tangent and a radius is 90 degrees
(c) \(120^\circ\) -
(a) \(56^\circ\)
(b) The two tangents drawn from a point to a circle are equal in length, so PA = PB -
(a) \(49^\circ\)
(b) The angle between a tangent and a chord is equal to the angle in the alternate segment -
(a) \(23^\circ\)
(b) The angle at the centre is twice the angle at the circumference -
(a) \(65^\circ\)
(b) Angles in the same segment are equal -
(a) \(90^\circ\)
(b) The angle in a semicircle is 90 degrees
(c) \(67^\circ\)