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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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Age band:Ages 13-15Ages 16-18

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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 = 134°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 134° · ∠ABC + ∠ADC = 180°

  2. 2. 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°

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

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

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 135°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 135° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 24° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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°
12 questions · Seed 51 · 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 = 134°, find the opposite interior angle ∠ADC.

    46°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 134° = 46°.

  2. 2. 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°.

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

    67°

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

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 135°, find the opposite interior angle ∠ADC.

    45°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 135° = 45°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.

    48°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 24° = 48°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    56°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 96° = 84°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 51 · 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 = 134°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 134° · ∠ABC + ∠ADC = 180°

  2. 2. 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°

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

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

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 135°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 135° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 24° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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°
12 questions · Seed 51 · 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 = 134°, find the opposite interior angle ∠ADC.

    46°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 134° = 46°.

  2. 2. 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°.

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

    67°

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

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 135°, find the opposite interior angle ∠ADC.

    45°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 135° = 45°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.

    48°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 24° = 48°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    56°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 96° = 84°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 51 · Age guidance, not curriculum certification