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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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 76°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 76° · ∠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. 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°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 68° · ∠ABC + ∠ADC = 180°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  9. 9. 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 = ?

  10. 10. 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 = ?

  11. 11. 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 = ?

  12. 12. 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°
12 questions · Seed 104742 · 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. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 76°, find the angle at the centre ∠AOB.

    152°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 76° = 152°.

  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. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    112°

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

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  9. 9. 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°.

  10. 10. 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°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 104742 · Age guidance, not curriculum certification
Worksheet StudioPractice pack · Ages 16-18

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 = 76°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 76° · ∠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. 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°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 62° · ∠AOB = 2 × ∠ACB = ?

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 68° · ∠ABC + ∠ADC = 180°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 80° · ∠AOB = 2 × ∠ACB = ?

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 71° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 48° · ∠AOB = 2 × ∠ACB = ?

  9. 9. 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 = ?

  10. 10. 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 = ?

  11. 11. 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 = ?

  12. 12. 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°
12 questions · Seed 104742 · Age guidance, not curriculum certification
Worksheet StudioAnswer key · Ages 16-18

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 = 76°, find the angle at the centre ∠AOB.

    152°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 76° = 152°.

  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. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 62°, find the angle at the centre ∠AOB.

    124°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 62° = 124°.

  5. 5. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 68°, find the opposite interior angle ∠ADC.

    112°

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

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 80°, find the angle at the centre ∠AOB.

    160°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 80° = 160°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 71°, find the angle at the centre ∠AOB.

    142°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 71° = 142°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 48°, find the angle at the centre ∠AOB.

    96°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 48° = 96°.

  9. 9. 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°.

  10. 10. 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°.

  11. 11. 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°.

  12. 12. 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°.

12 questions · Seed 104742 · Age guidance, not curriculum certification