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 = 77°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?
2. 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 = ?
3. 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 = ?
4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°
5. 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°
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 38° · ∠AOB = 2 × ∠ACB = ?
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?
10. 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 = ?
11. 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 = ?
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

12 questions · Seed 46 · 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 = 77°, find the angle at the centre ∠AOB.
154°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.
2. 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°.
3. 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°.
4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.
125°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.
5. 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°.
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.
76°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 38° = 76°.
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.
52°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.
86°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.
140°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.
10. 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°.
11. 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°.
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.
50°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

12 questions · Seed 46 · 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 = 77°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 77° · ∠AOB = 2 × ∠ACB = ?
2. 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 = ?
3. 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 = ?
4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 55° · ∠ABC + ∠ADC = 180°
5. 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°
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 38° · ∠AOB = 2 × ∠ACB = ?
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 26° · ∠AOB = 2 × ∠ACB = ?
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.
Cyclic Quad: ∠ABC = 94° · ∠ABC + ∠ADC = 180°
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 70° · ∠AOB = 2 × ∠ACB = ?
10. 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 = ?
11. 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 = ?
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.
Circle O: ∠ACB = 25° · ∠AOB = 2 × ∠ACB = ?

12 questions · Seed 46 · 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 = 77°, find the angle at the centre ∠AOB.
154°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 77° = 154°.
2. 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°.
3. 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°.
4. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 55°, find the opposite interior angle ∠ADC.
125°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 55° = 125°.
5. 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°.
6. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 38°, find the angle at the centre ∠AOB.
76°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 38° = 76°.
7. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 26°, find the angle at the centre ∠AOB.
52°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 26° = 52°.
8. ABCD is a cyclic quadrilateral inscribed in a circle. If interior angle ∠ABC = 94°, find the opposite interior angle ∠ADC.
86°
Opposite angles of a cyclic quadrilateral sum to 180°: 180° − 94° = 86°.
9. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 70°, find the angle at the centre ∠AOB.
140°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 70° = 140°.
10. 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°.
11. 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°.
12. Points A, B, and C lie on a circle with centre O. If the angle at the circumference ∠ACB = 25°, find the angle at the centre ∠AOB.
50°
By the circle theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc: 2 × 25° = 50°.

12 questions · Seed 46 · Age guidance, not curriculum certification