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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. 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°

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  3. 3. 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 = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 73°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 73° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 109°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 109° · ∠ABC + ∠ADC = 180°

  7. 7. 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°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 52°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 52° · ∠AOB = 2 × ∠ACB = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 102° · ∠ABC + ∠ADC = 180°

  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 = 74°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 74° · ∠ABC + ∠ADC = 180°
12 questions · Seed 53 · 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. 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°.

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.

    72°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 73°, find the angle at the centre ∠AOB.

    146°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 73° = 146°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 109°, find the opposite interior angle ∠ADC.

    71°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 109° = 71°.

  7. 7. 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°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 52°, find the angle at the centre ∠AOB.

    104°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 52° = 104°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    78°

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

  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 = 74°, find the opposite interior angle ∠ADC.

    106°

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

12 questions · Seed 53 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 13-15

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 = 78°, find the opposite interior angle ∠ADC.

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 55° · ∠AOB = 2 × ∠ACB = ?

  3. 3. 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 = ?

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 73°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 73° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 109°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 109° · ∠ABC + ∠ADC = 180°

  7. 7. 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°

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 52°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 52° · ∠AOB = 2 × ∠ACB = ?

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 102° · ∠ABC + ∠ADC = 180°

  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 = 74°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 74° · ∠ABC + ∠ADC = 180°
12 questions · Seed 53 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 13-15

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 = 78°, find the opposite interior angle ∠ADC.

    102°

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 55°, find the angle at the centre ∠AOB.

    110°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 55° = 110°.

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.

    72°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 73°, find the angle at the centre ∠AOB.

    146°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 73° = 146°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 109°, find the opposite interior angle ∠ADC.

    71°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 109° = 71°.

  7. 7. 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°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 52°, find the angle at the centre ∠AOB.

    104°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 52° = 104°.

  9. 9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 77°, find the angle at the centre ∠AOB.

    154°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 102°, find the opposite interior angle ∠ADC.

    78°

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

  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 = 74°, find the opposite interior angle ∠ADC.

    106°

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

12 questions · Seed 53 · Age guidance, not curriculum certification