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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 = 132°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 132° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 106° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 87° · ∠ABC + ∠ADC = 180°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠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. 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°

  8. 8. 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°

  9. 9. 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°

  10. 10. 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°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 93° · ∠ABC + ∠ADC = 180°
12 questions · Seed 44 · 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 = 132°, find the opposite interior angle ∠ADC.

    48°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 132° = 48°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    74°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 106° = 74°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    93°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 87° = 93°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  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. 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°.

  8. 8. 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°.

  9. 9. 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°.

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    87°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 93° = 87°.

12 questions · Seed 44 · 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 = 132°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 132° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 106° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 87° · ∠ABC + ∠ADC = 180°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠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. 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°

  8. 8. 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°

  9. 9. 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°

  10. 10. 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°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 93° · ∠ABC + ∠ADC = 180°
12 questions · Seed 44 · 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 = 132°, find the opposite interior angle ∠ADC.

    48°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 132° = 48°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    74°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 106° = 74°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    93°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 87° = 93°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  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. 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°.

  8. 8. 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°.

  9. 9. 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°.

  10. 10. 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°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    87°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 93° = 87°.

12 questions · Seed 44 · Age guidance, not curriculum certification