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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 = 49°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 82° · ∠AOB = 2 × ∠ACB = ?

  3. 3. 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°

  4. 4. 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 = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 68° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 75° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 62°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 62° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 98° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 59° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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 questions · Seed 43 · 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 = 49°, find the angle at the centre ∠AOB.

    98°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    164°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 82° = 164°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    136°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 68° = 136°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    105°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 75° = 105°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 62°, find the opposite interior angle ∠ADC.

    118°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 62° = 118°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    82°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 98° = 82°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    121°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 59° = 121°.

  11. 11. 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°.

  12. 12. 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 questions · Seed 43 · 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 = 49°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 82° · ∠AOB = 2 × ∠ACB = ?

  3. 3. 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°

  4. 4. 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 = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 68° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 75° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 62°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 62° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 98° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 59° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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 questions · Seed 43 · 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 = 49°, find the angle at the centre ∠AOB.

    98°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    164°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 82° = 164°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 68°, find the angle at the centre ∠AOB.

    136°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 68° = 136°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 75°, find the opposite interior angle ∠ADC.

    105°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 75° = 105°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 62°, find the opposite interior angle ∠ADC.

    118°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 62° = 118°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 98°, find the opposite interior angle ∠ADC.

    82°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 98° = 82°.

  9. 9. 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°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    121°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 59° = 121°.

  11. 11. 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°.

  12. 12. 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 questions · Seed 43 · Age guidance, not curriculum certification