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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 = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 119° · ∠ABC + ∠ADC = 180°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 98° · ∠ABC + ∠ADC = 180°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 108° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 107°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 107° · ∠ABC + ∠ADC = 180°

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 67° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104738 · 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 = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.

    61°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 119° = 61°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    82°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 98° = 82°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

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

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    72°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 108° = 72°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 100° = 80°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 107°, find the opposite interior angle ∠ADC.

    73°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 107° = 73°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    134°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 67° = 134°.

12 questions · Seed 104738 · 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 = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 119° · ∠ABC + ∠ADC = 180°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 98° · ∠ABC + ∠ADC = 180°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 108° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 107°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 107° · ∠ABC + ∠ADC = 180°

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 67° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104738 · 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 = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.

    61°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 119° = 61°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    82°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 98° = 82°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

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

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    72°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 108° = 72°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 100° = 80°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 107°, find the opposite interior angle ∠ADC.

    73°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 107° = 73°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    134°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 67° = 134°.

12 questions · Seed 104738 · Age guidance, not curriculum certification