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MathsAges 13-15

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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Worksheet StudioPractice pack · Ages 13-15

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 = 113°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 113° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. 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°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 129° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 58° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 81° · ∠ABC + ∠ADC = 180°
12 questions · Seed 56 · Age guidance, not curriculum certification
View the answer key
Worksheet StudioAnswer key · Ages 13-15

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 = 113°, find the opposite interior angle ∠ADC.

    67°

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

  2. 2. 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°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    51°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 129° = 51°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    86°

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

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    116°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 58° = 116°.

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

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

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    99°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 81° = 99°.

12 questions · Seed 56 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 13-15

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 = 113°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 113° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. 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°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 129° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 58° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 81° · ∠ABC + ∠ADC = 180°
12 questions · Seed 56 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 13-15

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 = 113°, find the opposite interior angle ∠ADC.

    67°

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

  2. 2. 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°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    51°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 129° = 51°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    86°

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

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    116°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 58° = 116°.

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

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

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    99°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 81° = 99°.

12 questions · Seed 56 · Age guidance, not curriculum certification