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MathsAges 13-15

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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Worksheet StudioPractice pack · Ages 13-15

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 = 35°, find the angle at the centre ∠AOB.

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

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 85°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 85° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 134°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 134° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 125° · ∠ABC + ∠ADC = 180°

  8. 8. 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°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 64°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 64° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 81° · ∠ABC + ∠ADC = 180°
12 questions · Seed 54 · Age guidance, not curriculum certification
View the answer key
Worksheet StudioAnswer key · Ages 13-15

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 = 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°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 85°, find the opposite interior angle ∠ADC.

    95°

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

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    115°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 134°, find the opposite interior angle ∠ADC.

    46°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 134° = 46°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.

    55°

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

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 64°, find the opposite interior angle ∠ADC.

    116°

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

  10. 10. 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°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    68°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    99°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 81° = 99°.

12 questions · Seed 54 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 13-15

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 = 35°, find the angle at the centre ∠AOB.

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

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 85°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 85° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 134°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 134° · ∠ABC + ∠ADC = 180°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 125° · ∠ABC + ∠ADC = 180°

  8. 8. 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°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 64°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 64° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 81° · ∠ABC + ∠ADC = 180°
12 questions · Seed 54 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 13-15

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 = 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°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 85°, find the opposite interior angle ∠ADC.

    95°

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

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.

    115°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 134°, find the opposite interior angle ∠ADC.

    46°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 134° = 46°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.

    55°

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

  8. 8. 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°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 64°, find the opposite interior angle ∠ADC.

    116°

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

  10. 10. 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°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.

    68°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 81°, find the opposite interior angle ∠ADC.

    99°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 81° = 99°.

12 questions · Seed 54 · Age guidance, not curriculum certification