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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 = 73°, find the opposite interior angle ∠ADC.

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 59° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 93° · ∠ABC + ∠ADC = 180°

  4. 4. 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°

  5. 5. 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 = ?

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 58° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 80° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 132°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 132° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104731 · 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 = 73°, find the opposite interior angle ∠ADC.

    107°

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    118°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 59° = 118°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    87°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 93° = 87°.

  4. 4. 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°.

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    156°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 78° = 156°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    116°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 58° = 116°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    100°

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

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 132°, find the opposite interior angle ∠ADC.

    48°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 132° = 48°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

12 questions · Seed 104731 · 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 = 73°, find the opposite interior angle ∠ADC.

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 59° · ∠AOB = 2 × ∠ACB = ?

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 93° · ∠ABC + ∠ADC = 180°

  4. 4. 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°

  5. 5. 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 = ?

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 78° · ∠AOB = 2 × ∠ACB = ?

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 58° · ∠AOB = 2 × ∠ACB = ?

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 72° · ∠AOB = 2 × ∠ACB = ?

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 80° · ∠ABC + ∠ADC = 180°

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 104° · ∠ABC + ∠ADC = 180°

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 132°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 132° · ∠ABC + ∠ADC = 180°

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 127° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104731 · 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 = 73°, find the opposite interior angle ∠ADC.

    107°

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

  2. 2. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 59°, find the angle at the centre ∠AOB.

    118°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 59° = 118°.

  3. 3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 93°, find the opposite interior angle ∠ADC.

    87°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 93° = 87°.

  4. 4. 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°.

  5. 5. 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°.

  6. 6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 78°, find the angle at the centre ∠AOB.

    156°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 78° = 156°.

  7. 7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 58°, find the angle at the centre ∠AOB.

    116°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 58° = 116°.

  8. 8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 72°, find the angle at the centre ∠AOB.

    144°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 72° = 144°.

  9. 9. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 80°, find the opposite interior angle ∠ADC.

    100°

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

  10. 10. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 104°, find the opposite interior angle ∠ADC.

    76°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 104° = 76°.

  11. 11. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 132°, find the opposite interior angle ∠ADC.

    48°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 132° = 48°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 127°, find the opposite interior angle ∠ADC.

    53°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 127° = 53°.

12 questions · Seed 104731 · Age guidance, not curriculum certification