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 13-15 Ages 16-18

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Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

1. 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 = ?
2. 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 = ?
3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 65° · ∠AOB = 2 × ∠ACB = ?
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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 119° · ∠ABC + ∠ADC = 180°
6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 125° · ∠ABC + ∠ADC = 180°
7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 112° · ∠ABC + ∠ADC = 180°
8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 24° · ∠AOB = 2 × ∠ACB = ?
9. 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 = ?
10. 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 = ?
11. 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 = ?
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

12 questions · Seed 104732 · 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. 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°.
2. 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°.
3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.
130°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 65° = 130°.
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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.
61°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 119° = 61°.
6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.
55°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 125° = 55°.
7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.
68°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 112° = 68°.
8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.
48°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 24° = 48°.
9. 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°.
10. 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°.
11. 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°.
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.
68°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Circle theorems & cyclic quadrilaterals

Take your time. Show your thinking.

1. 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 = ?
2. 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 = ?
3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 65° · ∠AOB = 2 × ∠ACB = ?
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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 119° · ∠ABC + ∠ADC = 180°
6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 125° · ∠ABC + ∠ADC = 180°
7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 112° · ∠ABC + ∠ADC = 180°
8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 24° · ∠AOB = 2 × ∠ACB = ?
9. 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 = ?
10. 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 = ?
11. 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 = ?
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 34° · ∠AOB = 2 × ∠ACB = ?

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Circle theorems & cyclic quadrilaterals

Answers and reasoning. Review before teaching.

1. 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°.
2. 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°.
3. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 65°, find the angle at the centre ∠AOB.
130°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 65° = 130°.
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. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 119°, find the opposite interior angle ∠ADC.
61°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 119° = 61°.
6. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 125°, find the opposite interior angle ∠ADC.
55°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 125° = 55°.
7. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 112°, find the opposite interior angle ∠ADC.
68°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 112° = 68°.
8. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 24°, find the angle at the centre ∠AOB.
48°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 24° = 48°.
9. 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°.
10. 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°.
11. 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°.
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 34°, find the angle at the centre ∠AOB.
68°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 34° = 68°.

12 questions · Seed 104732 · Age guidance, not curriculum certification