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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 = 122°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 122° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 64° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 92° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 32°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 32° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 43° · ∠AOB = 2 × ∠ACB = ?

  12. 12. 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°
12 questions · Seed 55 · 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 = 122°, find the opposite interior angle ∠ADC.

    58°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 122° = 58°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 69° = 111°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    128°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 64° = 128°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    88°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 92° = 88°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

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

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

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

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    52°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 32°, find the angle at the centre ∠AOB.

    64°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 32° = 64°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    86°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 43° = 86°.

  12. 12. 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°.

12 questions · Seed 55 · 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 = 122°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 122° · ∠ABC + ∠ADC = 180°

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 69° · ∠ABC + ∠ADC = 180°

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 64° · ∠AOB = 2 × ∠ACB = ?

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 92° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 96° · ∠ABC + ∠ADC = 180°

  7. 7. 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 = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 116° · ∠ABC + ∠ADC = 180°

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 32°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 32° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 43° · ∠AOB = 2 × ∠ACB = ?

  12. 12. 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°
12 questions · Seed 55 · 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 = 122°, find the opposite interior angle ∠ADC.

    58°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 122° = 58°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 69°, find the opposite interior angle ∠ADC.

    111°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 69° = 111°.

  3. 3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 64°, find the angle at the centre ∠AOB.

    128°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 64° = 128°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 92°, find the opposite interior angle ∠ADC.

    88°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 92° = 88°.

  5. 5. 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°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 96°, find the opposite interior angle ∠ADC.

    84°

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

  7. 7. 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°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 116°, find the opposite interior angle ∠ADC.

    64°

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

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.

    52°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 32°, find the angle at the centre ∠AOB.

    64°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 32° = 64°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 43°, find the angle at the centre ∠AOB.

    86°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 43° = 86°.

  12. 12. 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°.

12 questions · Seed 55 · Age guidance, not curriculum certification