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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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 23° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 38° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 79° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 110° · ∠ABC + ∠ADC = 180°

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  12. 12. 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 = ?
12 questions · Seed 45 · 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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    115°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    46°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 23° = 46°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.

    76°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 38° = 76°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    52°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.

  6. 6. 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°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    101°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 79° = 101°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    70°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 110° = 70°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  12. 12. 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°.

12 questions · Seed 45 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

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 = 65°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 23° · ∠AOB = 2 × ∠ACB = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 38° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 79° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 110° · ∠ABC + ∠ADC = 180°

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  12. 12. 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 = ?
12 questions · Seed 45 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

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 = 65°, find the opposite interior angle ∠ADC.

    115°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 23°, find the angle at the centre ∠AOB.

    46°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 23° = 46°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.

    76°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 38° = 76°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    52°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.

  6. 6. 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°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 79°, find the opposite interior angle ∠ADC.

    101°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 79° = 101°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    70°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 110° = 70°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  12. 12. 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°.

12 questions · Seed 45 · Age guidance, not curriculum certification