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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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Age band:Ages 13-15Ages 16-18

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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 = 63°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 63° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°

  3. 3. 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°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 108° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 70° · ∠ABC + ∠ADC = 180°

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 82° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  12. 12. 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°
12 questions · Seed 104736 · 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 = 63°, find the angle at the centre ∠AOB.

    126°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 63° = 126°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 100° = 80°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    72°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 108° = 72°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    110°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 70° = 110°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    164°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 82° = 164°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  12. 12. 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°.

12 questions · Seed 104736 · 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 = 63°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 63° · ∠AOB = 2 × ∠ACB = ?

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 84° · ∠ABC + ∠ADC = 180°

  3. 3. 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°

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 69° · ∠AOB = 2 × ∠ACB = ?

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 100° · ∠ABC + ∠ADC = 180°

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 50° · ∠AOB = 2 × ∠ACB = ?

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 108° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 70° · ∠ABC + ∠ADC = 180°

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 82° · ∠AOB = 2 × ∠ACB = ?

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 56° · ∠ABC + ∠ADC = 180°

  12. 12. 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°
12 questions · Seed 104736 · 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 = 63°, find the angle at the centre ∠AOB.

    126°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 63° = 126°.

  2. 2. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 84°, find the opposite interior angle ∠ADC.

    96°

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

  3. 3. 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°.

  4. 4. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 69°, find the angle at the centre ∠AOB.

    138°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 69° = 138°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.

    140°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.

  6. 6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 100°, find the opposite interior angle ∠ADC.

    80°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 100° = 80°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 50°, find the angle at the centre ∠AOB.

    100°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 50° = 100°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 108°, find the opposite interior angle ∠ADC.

    72°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 108° = 72°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 70°, find the opposite interior angle ∠ADC.

    110°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 70° = 110°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 82°, find the angle at the centre ∠AOB.

    164°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 82° = 164°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 56°, find the opposite interior angle ∠ADC.

    124°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 56° = 124°.

  12. 12. 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°.

12 questions · Seed 104736 · Age guidance, not curriculum certification