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 = 49°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?
2. 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 = ?
3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°
4. 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°
5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 56° · ∠AOB = 2 × ∠ACB = ?
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 42° · ∠AOB = 2 × ∠ACB = ?
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?
10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?
11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?
12. 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 questions · Seed 104737 · 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 = 49°, find the angle at the centre ∠AOB.
98°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.
2. 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°.
3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.
115°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.
4. 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°.
5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.
112°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 56° = 112°.
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.
56°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.
84°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 42° = 84°.
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.
114°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.
72°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.
10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.
60°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.
11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.
120°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.
12. 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 questions · Seed 104737 · 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 = 49°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 49° · ∠AOB = 2 × ∠ACB = ?
2. 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 = ?
3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 65° · ∠ABC + ∠ADC = 180°
4. 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°
5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 56° · ∠AOB = 2 × ∠ACB = ?
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 28° · ∠AOB = 2 × ∠ACB = ?
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 42° · ∠AOB = 2 × ∠ACB = ?
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 66° · ∠ABC + ∠ADC = 180°
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 36° · ∠AOB = 2 × ∠ACB = ?
10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 30° · ∠AOB = 2 × ∠ACB = ?
11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 60° · ∠AOB = 2 × ∠ACB = ?
12. 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 questions · Seed 104737 · 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 = 49°, find the angle at the centre ∠AOB.
98°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 49° = 98°.
2. 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°.
3. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 65°, find the opposite interior angle ∠ADC.
115°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 65° = 115°.
4. 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°.
5. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 56°, find the angle at the centre ∠AOB.
112°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 56° = 112°.
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 28°, find the angle at the centre ∠AOB.
56°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 28° = 56°.
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 42°, find the angle at the centre ∠AOB.
84°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 42° = 84°.
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 66°, find the opposite interior angle ∠ADC.
114°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 66° = 114°.
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 36°, find the angle at the centre ∠AOB.
72°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 36° = 72°.
10. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 30°, find the angle at the centre ∠AOB.
60°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 30° = 60°.
11. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 60°, find the angle at the centre ∠AOB.
120°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 60° = 120°.
12. 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 questions · Seed 104737 · Age guidance, not curriculum certification