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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. 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°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 22°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 22° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  8. 8. 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°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 37°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 37° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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 = ?
12 questions · Seed 42 · 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. 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 22°, find the angle at the centre ∠AOB.

    44°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 22° = 44°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 37°, find the angle at the centre ∠AOB.

    74°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 37° = 74°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 42 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

  1. 1. 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°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 22°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 22° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 89° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 75° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°

  8. 8. 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°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 37°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 37° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°

  11. 11. 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°

  12. 12. 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 = ?
12 questions · Seed 42 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

  1. 1. 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 22°, find the angle at the centre ∠AOB.

    44°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 22° = 44°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 89°, find the opposite interior angle ∠ADC.

    91°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 75°, find the angle at the centre ∠AOB.

    150°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 75° = 150°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.

    125°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 37°, find the angle at the centre ∠AOB.

    74°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 37° = 74°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 42 · Age guidance, not curriculum certification