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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 = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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 = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 33°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 33° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 91° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 95°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 95° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 87° · ∠ABC + ∠ADC = 180°

  10. 10. 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°

  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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 77°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 77° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104740 · 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 = 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°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 33°, find the angle at the centre ∠AOB.

    66°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 33° = 66°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 116° = 64°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    89°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 91° = 89°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    60°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 95°, find the opposite interior angle ∠ADC.

    85°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 95° = 85°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    93°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 87° = 93°.

  10. 10. 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°.

  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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 77°, find the opposite interior angle ∠ADC.

    103°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 77° = 103°.

12 questions · Seed 104740 · 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 = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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 = ?

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 33°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 33° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 91° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 95°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 95° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 87° · ∠ABC + ∠ADC = 180°

  10. 10. 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°

  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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 77°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 77° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104740 · 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 = 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°.

  2. 2. 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°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 33°, find the angle at the centre ∠AOB.

    66°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 33° = 66°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 116° = 64°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 91°, find the opposite interior angle ∠ADC.

    89°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 91° = 89°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.

    60°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 95°, find the opposite interior angle ∠ADC.

    85°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 95° = 85°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 87°, find the opposite interior angle ∠ADC.

    93°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 87° = 93°.

  10. 10. 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°.

  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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 77°, find the opposite interior angle ∠ADC.

    103°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 77° = 103°.

12 questions · Seed 104740 · Age guidance, not curriculum certification