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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 = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 122°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 122° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 47°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 47° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 81° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 57° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 111°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 111° · ∠ABC + ∠ADC = 180°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 44°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 44° · ∠AOB = 2 × ∠ACB = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 86° · ∠ABC + ∠ADC = 180°

  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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 42° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 52 · 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 = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 122°, find the opposite interior angle ∠ADC.

    58°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 122° = 58°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 47°, find the angle at the centre ∠AOB.

    94°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 47° = 94°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    162°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 81° = 162°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    123°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 57° = 123°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 111°, find the opposite interior angle ∠ADC.

    69°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 111° = 69°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 44°, find the angle at the centre ∠AOB.

    88°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 44° = 88°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 116° = 64°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    94°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 86° = 94°.

  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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.

    84°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 42° = 84°.

12 questions · Seed 52 · 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 = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 122°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 122° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 47°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 47° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 81° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 57° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 111°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 111° · ∠ABC + ∠ADC = 180°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 44°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 44° · ∠AOB = 2 × ∠ACB = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 86° · ∠ABC + ∠ADC = 180°

  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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 42° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 52 · 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 = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 122°, find the opposite interior angle ∠ADC.

    58°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 122° = 58°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 47°, find the angle at the centre ∠AOB.

    94°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 47° = 94°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    162°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 81° = 162°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    123°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 57° = 123°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 111°, find the opposite interior angle ∠ADC.

    69°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 111° = 69°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 44°, find the angle at the centre ∠AOB.

    88°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 44° = 88°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 116° = 64°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    94°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 86° = 94°.

  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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.

    84°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 42° = 84°.

12 questions · Seed 52 · Age guidance, not curriculum certification