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MathsAges 13-15

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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Worksheet StudioPractice pack · Ages 13-15

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 = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 99° · ∠ABC + ∠ADC = 180°

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 118°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 118° · ∠ABC + ∠ADC = 180°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 90° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 103°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 103° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°
12 questions · Seed 48 · Age guidance, not curriculum certification
View the answer key
Worksheet StudioAnswer key · Ages 13-15

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 = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 89° = 91°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    81°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 99° = 81°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 118°, find the opposite interior angle ∠ADC.

    62°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 118° = 62°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    68°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    90°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 90° = 90°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 103°, find the opposite interior angle ∠ADC.

    77°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 103° = 77°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

12 questions · Seed 48 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 13-15

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 = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 99° · ∠ABC + ∠ADC = 180°

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 118°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 118° · ∠ABC + ∠ADC = 180°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 90° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 103°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 103° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°
12 questions · Seed 48 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 13-15

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 = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 89° = 91°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    81°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 99° = 81°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 118°, find the opposite interior angle ∠ADC.

    62°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 118° = 62°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    68°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    90°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 90° = 90°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 103°, find the opposite interior angle ∠ADC.

    77°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 103° = 77°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

12 questions · Seed 48 · Age guidance, not curriculum certification