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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 = 124°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 124° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 75° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 74° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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 = ?

  7. 7. 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°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 61° · ∠ABC + ∠ADC = 180°

  9. 9. 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°

  10. 10. 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 = ?

  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 = 64°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 64° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104743 · 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 = 124°, find the opposite interior angle ∠ADC.

    56°

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

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    105°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 75° = 105°.

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    148°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 74° = 148°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    119°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 61° = 119°.

  9. 9. 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°.

  10. 10. 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°.

  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 = 64°, find the opposite interior angle ∠ADC.

    116°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 64° = 116°.

12 questions · Seed 104743 · 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 = 124°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 124° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 75° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 74° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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 = ?

  7. 7. 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°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 61° · ∠ABC + ∠ADC = 180°

  9. 9. 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°

  10. 10. 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 = ?

  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 = 64°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 64° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104743 · 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 = 124°, find the opposite interior angle ∠ADC.

    56°

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

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    105°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 75° = 105°.

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    148°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 74° = 148°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 61°, find the opposite interior angle ∠ADC.

    119°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 61° = 119°.

  9. 9. 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°.

  10. 10. 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°.

  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 = 64°, find the opposite interior angle ∠ADC.

    116°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 64° = 116°.

12 questions · Seed 104743 · Age guidance, not curriculum certification