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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 = 76°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 76° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 128°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 128° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  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. 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 = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 120° · ∠ABC + ∠ADC = 180°
12 questions · Seed 58 · 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 = 76°, find the opposite interior angle ∠ADC.

    104°

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

  2. 2. 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°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

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

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 128°, find the opposite interior angle ∠ADC.

    52°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 128° = 52°.

  9. 9. 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°.

  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. 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°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    60°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 120° = 60°.

12 questions · Seed 58 · 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 = 76°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 76° · ∠ABC + ∠ADC = 180°

  2. 2. 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 = ?

  3. 3. 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 = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°

  5. 5. 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 = ?

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 128°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 128° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  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. 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 = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 120° · ∠ABC + ∠ADC = 180°
12 questions · Seed 58 · 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 = 76°, find the opposite interior angle ∠ADC.

    104°

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

  2. 2. 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°.

  3. 3. 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°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

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

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 128°, find the opposite interior angle ∠ADC.

    52°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 128° = 52°.

  9. 9. 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°.

  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. 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°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 120°, find the opposite interior angle ∠ADC.

    60°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 120° = 60°.

12 questions · Seed 58 · Age guidance, not curriculum certification