Small practice. Bright discoveries.Explore the free starter collection
worksheetstudio.
Sign in
Back to worksheets
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.

New variationTry interactive challengeBrowse templates
Age band:Ages 13-15Ages 16-18

Choose “Save as PDF” to keep a copy. Use your browser’s paper setting for A4 or Letter.

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 = 68°, find the angle at the centre ∠AOB.

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 78°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 78° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 53°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 53° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 110° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  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. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°
12 questions · Seed 49 · 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 = 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 78°, find the opposite interior angle ∠ADC.

    102°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 53°, find the angle at the centre ∠AOB.

    106°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 53° = 106°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    70°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 110° = 70°.

  6. 6. 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°.

  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. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

12 questions · Seed 49 · 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 = 68°, find the angle at the centre ∠AOB.

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 78°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 78° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 53°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 53° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 110° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  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. 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 = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°
12 questions · Seed 49 · 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 = 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.

    120°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 78°, find the opposite interior angle ∠ADC.

    102°

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

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 53°, find the angle at the centre ∠AOB.

    106°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 53° = 106°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 110°, find the opposite interior angle ∠ADC.

    70°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 110° = 70°.

  6. 6. 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°.

  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. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

12 questions · Seed 49 · Age guidance, not curriculum certification