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MathsAges 16-18

Circle theorems & cyclic quadrilaterals

Apply circle theorems for the angle at the centre (2× circumference angle) and opposite angles in a cyclic quadrilateral. Review suitability for your learners before teaching.

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Age band:Ages 13-15Ages 16-18

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Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 63°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 63° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 68° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 120° · ∠ABC + ∠ADC = 180°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 68° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 124°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 124° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 79° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°
12 questions · Seed 59 · Age guidance, not curriculum certification
View the answer key
Worksheet StudioAnswer key · Ages 16-18

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 63°, find the angle at the centre ∠AOB.

    126°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 63° = 126°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    136°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 68° = 136°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    60°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 120° = 60°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    112°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 68° = 112°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 124°, find the opposite interior angle ∠ADC.

    56°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 124° = 56°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    101°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 79° = 101°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 83° = 97°.

12 questions · Seed 59 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 63°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 63° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 68° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 120° · ∠ABC + ∠ADC = 180°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 68° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 124°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 124° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 79° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°
12 questions · Seed 59 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 63°, find the angle at the centre ∠AOB.

    126°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 63° = 126°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    136°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 68° = 136°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    60°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 120° = 60°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    112°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 68° = 112°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 124°, find the opposite interior angle ∠ADC.

    56°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 124° = 56°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    101°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 79° = 101°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 83° = 97°.

12 questions · Seed 59 · Age guidance, not curriculum certification