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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 = 57°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 57° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 63°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 63° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 102° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 79°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 79° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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 = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 76°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 76° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°
12 questions · Seed 47 · 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 = 57°, find the angle at the centre ∠AOB.

    114°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 57° = 114°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 63°, find the opposite interior angle ∠ADC.

    117°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    56°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    78°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 102° = 78°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 79°, find the angle at the centre ∠AOB.

    158°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 79° = 158°.

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 76°, find the opposite interior angle ∠ADC.

    104°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 76° = 104°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 96° = 84°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    86°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.

12 questions · Seed 47 · 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 = 57°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 57° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 63°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 63° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 102° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 79°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 79° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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 = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 76°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 76° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 46° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°
12 questions · Seed 47 · 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 = 57°, find the angle at the centre ∠AOB.

    114°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 57° = 114°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 63°, find the opposite interior angle ∠ADC.

    117°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.

    56°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    78°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 102° = 78°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 79°, find the angle at the centre ∠AOB.

    158°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 79° = 158°.

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 76°, find the opposite interior angle ∠ADC.

    104°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 76° = 104°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 96° = 84°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 46°, find the angle at the centre ∠AOB.

    92°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 46° = 92°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.

    86°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.

12 questions · Seed 47 · Age guidance, not curriculum certification