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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 = 36°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 23° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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°

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 131°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 131° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 72°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 72° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 101°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 101° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 104739 · 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 = 36°, find the angle at the centre ∠AOB.

    72°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 121° = 59°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    46°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 23° = 46°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 131°, find the opposite interior angle ∠ADC.

    49°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 131° = 49°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    50°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 72°, find the opposite interior angle ∠ADC.

    108°

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

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 101°, find the opposite interior angle ∠ADC.

    79°

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

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  12. 12. 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°.

12 questions · Seed 104739 · 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 = 36°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 23° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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°

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 131°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 131° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 72°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 72° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 101°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 101° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 104739 · 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 = 36°, find the angle at the centre ∠AOB.

    72°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 121° = 59°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    46°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 23° = 46°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 131°, find the opposite interior angle ∠ADC.

    49°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 131° = 49°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    50°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 72°, find the opposite interior angle ∠ADC.

    108°

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

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 101°, find the opposite interior angle ∠ADC.

    79°

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

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  12. 12. 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°.

12 questions · Seed 104739 · Age guidance, not curriculum certification