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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 = 66°, find the angle at the centre ∠AOB.

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

  2. 2. 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°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 73°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 73° · ∠ABC + ∠ADC = 180°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 117°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 117° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 129° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 59° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 99° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. 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 = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 114° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104745 · 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 = 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°.

  2. 2. 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°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 73°, find the opposite interior angle ∠ADC.

    107°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 73° = 107°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 117°, find the opposite interior angle ∠ADC.

    63°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 117° = 63°.

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    51°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 129° = 51°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    121°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 59° = 121°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    81°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 99° = 81°.

  10. 10. 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°.

  11. 11. 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°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    66°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 114° = 66°.

12 questions · Seed 104745 · 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 = 66°, find the angle at the centre ∠AOB.

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

  2. 2. 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°

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 73°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 73° · ∠ABC + ∠ADC = 180°

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 117°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 117° · ∠ABC + ∠ADC = 180°

  5. 5. 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°

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 35° · ∠AOB = 2 × ∠ACB = ?

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 129° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 59° · ∠ABC + ∠ADC = 180°

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 99° · ∠ABC + ∠ADC = 180°

  10. 10. 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 = ?

  11. 11. 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 = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 114° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104745 · 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 = 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°.

  2. 2. 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°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 73°, find the opposite interior angle ∠ADC.

    107°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 73° = 107°.

  4. 4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 117°, find the opposite interior angle ∠ADC.

    63°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 117° = 63°.

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 35°, find the angle at the centre ∠AOB.

    70°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 35° = 70°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 129°, find the opposite interior angle ∠ADC.

    51°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 129° = 51°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 59°, find the opposite interior angle ∠ADC.

    121°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 59° = 121°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 99°, find the opposite interior angle ∠ADC.

    81°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 99° = 81°.

  10. 10. 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°.

  11. 11. 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°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 114°, find the opposite interior angle ∠ADC.

    66°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 114° = 66°.

12 questions · Seed 104745 · Age guidance, not curriculum certification