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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 = 117°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 117° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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 = ?

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 80° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 70° · ∠ABC + ∠ADC = 180°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  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 = 44°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 44° · ∠AOB = 2 × ∠ACB = ?

  11. 11. 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. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 65° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104734 · 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 = 117°, find the opposite interior angle ∠ADC.

    63°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 117° = 63°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  4. 4. 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°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    100°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 80° = 100°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    110°

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

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  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 = 44°, find the angle at the centre ∠AOB.

    88°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 44° = 88°.

  11. 11. 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. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.

    130°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 65° = 130°.

12 questions · Seed 104734 · 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 = 117°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 117° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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 = ?

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 80° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 70° · ∠ABC + ∠ADC = 180°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  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 = 44°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 44° · ∠AOB = 2 × ∠ACB = ?

  11. 11. 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. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 65° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104734 · 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 = 117°, find the opposite interior angle ∠ADC.

    63°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 117° = 63°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  4. 4. 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°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    100°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 80° = 100°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    110°

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

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  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 = 44°, find the angle at the centre ∠AOB.

    88°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 44° = 88°.

  11. 11. 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. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.

    130°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 65° = 130°.

12 questions · Seed 104734 · Age guidance, not curriculum certification