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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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 64° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 57° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. 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 = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 97°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 97° · ∠ABC + ∠ADC = 180°

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 90° · ∠ABC + ∠ADC = 180°

  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 = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 115° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 104733 · 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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    128°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 64° = 128°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    123°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 57° = 123°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 97°, find the opposite interior angle ∠ADC.

    83°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 97° = 83°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    90°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 90° = 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 = 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°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    50°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    65°

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

  12. 12. 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°.

12 questions · Seed 104733 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 13-15

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 64° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 57° · ∠ABC + ∠ADC = 180°

  3. 3. 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 = ?

  4. 4. 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 = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 97°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 97° · ∠ABC + ∠ADC = 180°

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 90° · ∠ABC + ∠ADC = 180°

  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 = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 115° · ∠ABC + ∠ADC = 180°

  12. 12. 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 = ?
12 questions · Seed 104733 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 13-15

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

  1. 1. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    128°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 64° = 128°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 57°, find the opposite interior angle ∠ADC.

    123°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 57° = 123°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 97°, find the opposite interior angle ∠ADC.

    83°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 97° = 83°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 90°, find the opposite interior angle ∠ADC.

    90°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 90° = 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 = 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°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.

    50°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    65°

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

  12. 12. 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°.

12 questions · Seed 104733 · Age guidance, not curriculum certification