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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. 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°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 49°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 91° · ∠ABC + ∠ADC = 180°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 27° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 61° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 71°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 71° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  10. 10. 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 = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 67°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 67° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 50 · 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. 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 49°, find the angle at the centre ∠AOB.

    98°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    89°

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

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    54°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 27° = 54°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    119°

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

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 71°, find the opposite interior angle ∠ADC.

    109°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 71° = 109°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    156°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 78° = 156°.

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 67°, find the opposite interior angle ∠ADC.

    113°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 67° = 113°.

  12. 12. 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°.

12 questions · Seed 50 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

  1. 1. 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°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 49°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 91° · ∠ABC + ∠ADC = 180°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 27° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 61° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 71°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 71° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  10. 10. 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 = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 67°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 67° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 50 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

  1. 1. 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 49°, find the angle at the centre ∠AOB.

    98°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    89°

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

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    54°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 27° = 54°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    119°

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

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 71°, find the opposite interior angle ∠ADC.

    109°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 71° = 109°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    156°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 78° = 156°.

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 67°, find the opposite interior angle ∠ADC.

    113°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 67° = 113°.

  12. 12. 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°.

12 questions · Seed 50 · Age guidance, not curriculum certification