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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 = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  2. 2. 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°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 112° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 106° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 74° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 59° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 66°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 66° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 86° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104744 · 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 = 94°, find the opposite interior angle ∠ADC.

    86°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.

  2. 2. 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°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.

    68°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 112° = 68°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    74°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 106° = 74°.

  6. 6. 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°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    148°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 74° = 148°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    118°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 59° = 118°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 66°, find the angle at the centre ∠AOB.

    132°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 66° = 132°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    94°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 86° = 94°.

12 questions · Seed 104744 · 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 = 94°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°

  2. 2. 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°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 112° · ∠ABC + ∠ADC = 180°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 106° · ∠ABC + ∠ADC = 180°

  6. 6. 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°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 74° · ∠AOB = 2 × ∠ACB = ?

  8. 8. 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°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 59° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 66°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 66° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 45° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 86° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104744 · 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 = 94°, find the opposite interior angle ∠ADC.

    86°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.

  2. 2. 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°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.

    68°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 112° = 68°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 106°, find the opposite interior angle ∠ADC.

    74°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 106° = 74°.

  6. 6. 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°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 74°, find the angle at the centre ∠AOB.

    148°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 74° = 148°.

  8. 8. 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°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    118°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 59° = 118°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 66°, find the angle at the centre ∠AOB.

    132°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 66° = 132°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 45°, find the angle at the centre ∠AOB.

    90°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 45° = 90°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 86°, find the opposite interior angle ∠ADC.

    94°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 86° = 94°.

12 questions · Seed 104744 · Age guidance, not curriculum certification