Circle Theorems (Year 10)
Free printable UK Year 10 (GCSE Higher tier) maths worksheet on circle theorems: angle at the centre, angle in a semicircle, same segment, cyclic quadrilaterals and tangent-radius, with accurate diagrams.
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Grade 9 · Math worksheet
Name
Date
Math
Circle Theorems
Each diagram shows a circle with centre O (drawn accurately from the given values, but do not measure from it). Find each marked angle and NAME the circle theorem you used: the reason earns marks at GCSE, not just the number. Higher tier.
- 1.A, B and P lie on a circle with centre O. Angle AOB = 86° (at the centre). Find angle APB, marked x, and give a reason.
- 2.A, B and P lie on a circle with centre O. Angle APB = 26° (at the circumference). Find angle AOB, marked x, and give a reason.
- 3.A, B and P lie on a circle with centre O. Angle AOB = 62° (at the centre). Find angle APB, marked x, and give a reason.
- 4.AB is a diameter of the circle and P lies on the circle. Angle PAB = 52°. Find angle PBA, marked x, and give reasons.
- 5.AB is a diameter of the circle and P lies on the circle. Angle PAB = 33°. Find angle PBA, marked x, and give reasons.
- 6.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 58°. Find angle AQB, marked x, and give a reason.
- 7.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 38°. Find angle AQB, marked x, and give a reason.
- 8.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 55°. Find angle AQB, marked x, and give a reason.
- 9.ABCD is a cyclic quadrilateral. Angle A = 91° and angle B = 112°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 10.ABCD is a cyclic quadrilateral. Angle A = 66° and angle B = 81°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 11.ABCD is a cyclic quadrilateral. Angle A = 110° and angle B = 97°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 12.PT is a tangent to the circle with centre O, touching it at T. Angle OPT = 26°. Find angle TOP, marked x, and give reasons.
- 13.PT is a tangent to the circle with centre O, touching it at T. Angle OPT = 34°. Find angle TOP, marked x, and give reasons.
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