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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 = 66°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. 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°

  7. 7. 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 = ?

  8. 8. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 81° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 67° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104741 · 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 = 66°, find the opposite interior angle ∠ADC.

    114°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

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

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  5. 5. 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°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    162°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 81° = 162°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    134°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 67° = 134°.

12 questions · Seed 104741 · 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 = 66°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. 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°

  7. 7. 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 = ?

  8. 8. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 81° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 67° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104741 · 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 = 66°, find the opposite interior angle ∠ADC.

    114°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

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

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  5. 5. 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°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 81°, find the angle at the centre ∠AOB.

    162°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 81° = 162°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 67°, find the angle at the centre ∠AOB.

    134°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 67° = 134°.

12 questions · Seed 104741 · Age guidance, not curriculum certification