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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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  3. 3. 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°

  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 = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 92° · ∠ABC + ∠ADC = 180°

  8. 8. 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 = ?

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  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 104730 · 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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

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

  3. 3. 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°.

  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 = 94°, find the opposite interior angle ∠ADC.

    86°

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

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    88°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 92° = 88°.

  8. 8. 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°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

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

  11. 11. 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°.

  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 104730 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

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 = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  3. 3. 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°

  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 = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  6. 6. 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 = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 92° · ∠ABC + ∠ADC = 180°

  8. 8. 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 = ?

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 83° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  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 104730 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

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 = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

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

  3. 3. 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°.

  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 = 94°, find the opposite interior angle ∠ADC.

    86°

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

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    88°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 92° = 88°.

  8. 8. 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°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 83°, find the opposite interior angle ∠ADC.

    97°

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

  11. 11. 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°.

  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 104730 · Age guidance, not curriculum certification