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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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Age band:Ages 13-15Ages 16-18

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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 = 29°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 29° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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 = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 60°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 60° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 114° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 40°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 40° · ∠AOB = 2 × ∠ACB = ?

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 56° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104735 · 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 = 29°, find the angle at the centre ∠AOB.

    58°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 29° = 58°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 60°, find the opposite interior angle ∠ADC.

    120°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 60° = 120°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    66°

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

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 69° = 111°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 40°, find the angle at the centre ∠AOB.

    80°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 40° = 80°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.

    112°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 56° = 112°.

12 questions · Seed 104735 · 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 = 29°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 29° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  4. 4. 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 = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 60°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 60° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 114° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 40°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 40° · ∠AOB = 2 × ∠ACB = ?

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 56° · ∠AOB = 2 × ∠ACB = ?
12 questions · Seed 104735 · 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 = 29°, find the angle at the centre ∠AOB.

    58°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 29° = 58°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  4. 4. 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°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 60°, find the opposite interior angle ∠ADC.

    120°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 60° = 120°.

  6. 6. 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°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    66°

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

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 69° = 111°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 40°, find the angle at the centre ∠AOB.

    80°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 40° = 80°.

  12. 12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.

    112°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 56° = 112°.

12 questions · Seed 104735 · Age guidance, not curriculum certification