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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 = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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°

  3. 3. 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°

  4. 4. 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°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 115° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 57°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 57° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 27° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104729 · 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 = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  2. 2. 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°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    65°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 115° = 65°.

  9. 9. 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°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 57°, find the angle at the centre ∠AOB.

    114°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 57° = 114°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    54°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 27° = 54°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

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

12 questions · Seed 104729 · 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 = 54°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 54° · ∠AOB = 2 × ∠ACB = ?

  2. 2. 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°

  3. 3. 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°

  4. 4. 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°

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 61° · ∠AOB = 2 × ∠ACB = ?

  6. 6. 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°

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 133° · ∠ABC + ∠ADC = 180°

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 115° · ∠ABC + ∠ADC = 180°

  9. 9. 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 = ?

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 57°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 57° · ∠AOB = 2 × ∠ACB = ?

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    Circle O: ∠ACB = 27° · ∠AOB = 2 × ∠ACB = ?

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    Cyclic Quad: ∠ABC = 121° · ∠ABC + ∠ADC = 180°
12 questions · Seed 104729 · 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 = 54°, find the angle at the centre ∠AOB.

    108°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 54° = 108°.

  2. 2. 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°.

  3. 3. 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°.

  4. 4. 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°.

  5. 5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 61°, find the angle at the centre ∠AOB.

    122°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 61° = 122°.

  6. 6. 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°.

  7. 7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 133°, find the opposite interior angle ∠ADC.

    47°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 133° = 47°.

  8. 8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 115°, find the opposite interior angle ∠ADC.

    65°

    Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 115° = 65°.

  9. 9. 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°.

  10. 10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 57°, find the angle at the centre ∠AOB.

    114°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 57° = 114°.

  11. 11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 27°, find the angle at the centre ∠AOB.

    54°

    By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 27° = 54°.

  12. 12. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 121°, find the opposite interior angle ∠ADC.

    59°

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

12 questions · Seed 104729 · Age guidance, not curriculum certification