Circle Theorems – Grade 9 IGCSE Maths

Theorem 1

Angle in a semicircle = 90°

Theorem 2

Angles in same segment are equal

Theorem 3

Angle at centre = 2 × angle at circumference

Theorem 4

Cyclic quadrilateral: opposite angles sum to 180°

Theorems 5–6

Tangent–radius perpendicular, equal tangents

Theorem 7

Alternate segment theorem

Key Vocabulary

Chord: A line segment with both endpoints on the circle.
Arc: Part of the circumference between two points.
Tangent: A line that touches the circle at exactly one point.
Cyclic polygon: All vertices lie on the circle. A cyclic quadrilateral has four vertices on the circle.

Theorem 1 — Angle in a Semicircle

The angle in a semicircle is always 90°

If AB is a diameter, then angle ACB = 90° for any point C on the circle (in the major arc). This is also called Thales' theorem.

Theorem 2 — Angles in the Same Segment

Angles subtended by the same arc in the same segment are equal

If A, B, C, D are on the circle and all on the same side of chord PQ, then angle PAQ = angle PBQ = angle PCQ...

Theorem 3 — Angle at the Centre

Angle at centre = 2 × angle at circumference (same arc)

The central angle (subtended at the centre O) is twice the inscribed angle (subtended at any point on the circumference), both using the same arc.
Example: If angle at circumference = 35°, then angle at centre = 70°.

Theorem 4 — Cyclic Quadrilateral

Opposite angles of a cyclic quadrilateral sum to 180°

In a cyclic quadrilateral ABCD: angle A + angle C = 180° and angle B + angle D = 180°.
This is because opposite angles are subtended by supplementary arcs.

Theorem 5 — Tangent and Radius

A tangent to a circle is perpendicular to the radius at the point of contact

If T is the point where a tangent meets the circle, and O is the centre, then angle OTP = 90°.

Theorem 6 — Two Tangents from an External Point

Tangents from an external point are equal in length

If two tangents are drawn from external point P to touch the circle at A and B, then PA = PB. Also, the line PO bisects angle APB and angle AOB.

Theorem 7 — Alternate Segment Theorem

Angle between tangent and chord = angle in the alternate segment

The angle between a tangent at a point and a chord drawn from that point equals the inscribed angle subtending the chord from the other side.
Example: If tangent-chord angle at T = 65°, then the angle in the alternate segment = 65°.

The "alternate" segment is the segment on the OTHER side of the chord from where the tangent meets the circle.

Example 1 — Angle in Semicircle

AB is a diameter. Angle BAC = 35°. Find angle ABC.

Theorem 1: Angle ACB = 90° (angle in semicircle)

Angle sum: Angle ABC = 180° − 90° − 35° = 55°

Example 2 — Angle at Centre

Angle at circumference = 40°. Find the angle at the centre (same arc).

Theorem 3: Angle at centre = 2 × angle at circumference

Answer: 2 × 40° = 80°

Example 3 — Cyclic Quadrilateral

Cyclic quadrilateral ABCD. Angle A = 65°. Find angle C.

Theorem 4: Opposite angles sum to 180°

Answer: Angle C = 180° − 65° = 115°

Example 4 — Tangent–Radius

A tangent meets the circle at T. Angle between the tangent and a line from T to another point on the circle = 40°. The radius OT = 5 cm, and the tangent length from external point = 12 cm. Find the angle between radius and tangent.

Theorem 5: The tangent is perpendicular to the radius at T.

Answer: Angle OTP = 90°

Example 5 — Alternate Segment

The angle between a tangent at T and chord TP = 65°. Find the angle in the alternate segment (angle TQP where Q is on the major arc).

Theorem 7: Angle in alternate segment = angle between tangent and chord

Answer: Angle TQP = 65°

Example 6 — Combined Theorems

O is the centre. Angle AOB = 130° (reflex). Find the angle at the circumference on the major arc.

Reflex angle at centre: 360° − 130° = 230°

Theorem 3: Angle at circumference (minor arc side) = 130° ÷ 2 = 65°

⭕ Circle Theorem Diagrams

Select a theorem to see an annotated diagram with example angles.

Theorem:

Select a theorem and click Show Diagram.

Exercise 1 — Angle in a Semicircle

AB is a diameter. Find the missing angle in each triangle inscribed in the semicircle.

1. Angle ACB = ? (C is on circle, AB is diameter)

2. AB is diameter. Angle BAC = 35°. Find angle ABC.

3. AB is diameter. Angle BAC = 48°. Find angle ABC.

4. AB is diameter. Angle ABC = 17°. Find angle BAC.

5. AB is diameter. Angle BAC = 52°. Find angle ABC.

6. AB is diameter. Angle ABC = 29°. Find angle BAC.

7. AB is diameter. Angle BAC = 61°. Find angle ABC.

8. AB is diameter. Angle BAC = 9°. Find angle ABC.

9. AB is diameter. Angle ABC = 43°. Find angle BAC.

10. AB is diameter. Angle BAC = 24°. Find angle ABC.

Exercise 2 — Angle at Centre (find the angle at centre)

1. Angle at circumference = 25°. Angle at centre?

2. Angle at circumference = 40°. Angle at centre?

3. Angle at circumference = 55°. Angle at centre?

4. Angle at circumference = 70°. Angle at centre?

5. Angle at circumference = 35°. Angle at centre?

6. Angle at circumference = 50°. Angle at centre?

7. Angle at circumference = 65°. Angle at centre?

8. Angle at circumference = 30°. Angle at centre?

9. Angle at circumference = 45°. Angle at centre?

10. Angle at circumference = 60°. Angle at centre?

Exercise 3 — Cyclic Quadrilateral (find the missing angle)

1. Cyclic quad ABCD. Angle A = 65°. Find angle C.

2. Cyclic quad. Angle B = 85°. Find opposite angle D.

3. Cyclic quad. Angle A = 72°. Find angle C.

4. Cyclic quad. Angle D = 93°. Find angle B.

5. Cyclic quad. Angle A = 57°. Find angle C.

6. Cyclic quad. Angle B = 106°. Find angle D.

7. Cyclic quad. Angle A = 78°. Find angle C.

8. Cyclic quad. Angle C = 89°. Find angle A.

9. Cyclic quad. Angle B = 42°. Find angle D.

10. Cyclic quad. Angle A = 117°. Find angle C.

Exercise 4 — Tangent–Radius Problems (find the angle)

Use the fact that a tangent meets the radius at 90°. Give angle OTP or solve using the triangle formed.

1. Tangent meets circle at T. Radius OT. Angle between tangent and radius?

2. OT is radius, TP is tangent. Angle TOP = 90°. Angle OPT?

3. From external point P, tangent to T. Angle OPT = 40°. Find angle TOP.

4. Angle between two tangents from P to circle = 80°. Find angle at P.

5. Tangent-radius angle = 90°. Angle OTP = ?

6. From external P. Angle OPT = 25°. Find angle TOP (triangle OTP).

7. From external P. Angle OPT = 18°. Find angle TOP.

8. From external P. Angle OPT = 32°. Find angle TOP.

9. From external P. Angle OPT = 9°. Find angle TOP.

10. From external P. Angle OPT = 46°. Find angle TOP.

Exercise 5 — Alternate Segment Theorem

1. Tangent-chord angle at T = 65°. Angle in alternate segment?

2. Tangent-chord angle at T = 40°. Angle in alternate segment?

3. Angle in alternate segment = 78°. Tangent-chord angle?

4. Tangent-chord angle = 52°. Angle in alternate segment?

5. Tangent-chord angle = 83°. Angle in alternate segment?

6. Angle in alternate segment = 37°. Tangent-chord angle?

7. Tangent-chord angle = 61°. Angle in alternate segment?

8. Tangent-chord angle = 49°. Angle in alternate segment?

9. Angle in alternate segment = 74°. Tangent-chord angle?

10. Tangent-chord angle = 56°. Angle in alternate segment?

🏋️ Practice — 20 Questions

1. Angle in semicircle = ?

2. AB diam. Angle BAC=35°. Find angle ABC.

3. AB diam. Angle BAC=48°. Find angle ABC.

4. AB diam. Angle ABC=17°. Find angle BAC.

5. AB diam. Angle BAC=52°. Find angle ABC.

6. AB diam. Angle ABC=29°. Find angle BAC.

7. AB diam. Angle BAC=61°. Find angle ABC.

8. AB diam. Angle BAC=9°. Find angle ABC.

9. AB diam. Angle ABC=43°. Find angle BAC.

10. AB diam. Angle BAC=24°. Find angle ABC.

11. Angle at circumference=25°. Angle at centre?

12. Angle at circumference=40°. Angle at centre?

13. Angle at circumference=55°. Angle at centre?

14. Angle at circumference=70°. Angle at centre?

15. Angle at circumference=35°. Angle at centre?

16. Angle at circumference=50°. Angle at centre?

17. Angle at circumference=65°. Angle at centre?

18. Angle at circumference=30°. Angle at centre?

19. Angle at circumference=45°. Angle at centre?

20. Angle at circumference=60°. Angle at centre?

🏆 Challenge — 8 Questions

1. Cyclic quad. Angle A=65°. Find angle C.

2. Cyclic quad. Angle B=85°. Find angle D.

3. Cyclic quad. Angle A=72°. Find angle C.

4. Cyclic quad. Angle D=93°. Find angle B.

5. Cyclic quad. Angle A=57°. Find angle C.

6. Cyclic quad. Angle B=106°. Find angle D.

7. Cyclic quad. Angle A=78°. Find angle C.

8. Cyclic quad. Angle C=89°. Find angle A.

⭕ Circle Theorems