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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 = 39°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 39° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°

  8. 8. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 43° · ∠AOB = 2 × ∠ACB = ?

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 57 · 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 = 39°, find the angle at the centre ∠AOB.

    78°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 39° = 78°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    60°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 83° = 97°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.

    114°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    86°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 43° = 86°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

12 questions · Seed 57 · 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 = 39°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 39° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°

  8. 8. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 43° · ∠AOB = 2 × ∠ACB = ?

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 57 · 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 = 39°, find the angle at the centre ∠AOB.

    78°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 39° = 78°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    60°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 83° = 97°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.

    114°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    86°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 43° = 86°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

12 questions · Seed 57 · Age guidance, not curriculum certification