IGCSE Circle Theorems: Find Missing Angles with Reasons
Learn Cambridge IGCSE circle theorems through clear examples, exam-style explanations and original practice questions for Core and Extended.
By Math School Editorial Team
Why circle theorems feel difficult
Circle-theorem questions often contain very little arithmetic. The challenge is recognising which geometrical property applies and explaining the reasoning. A correct numerical answer without a clear justification may not demonstrate the skill the examiner is testing.
For Cambridge IGCSE Mathematics 0580 examinations in 2025–2027, Core topic C4.7 covers angles in a semicircle and the right angle between a tangent and radius. Extended E4.7 adds the centre–circumference theorem, angles in the same segment, cyclic quadrilaterals and the alternate segment theorem. Extended E4.8 adds chord and tangent symmetry properties. Requirements differ between exam boards, so always check the correct syllabus.
1. Angle in a semicircle
If AB is a diameter and C is another point on the circle, angle ACB is 90°.
Example: Angle CAB = 37°. Find angle ABC.
Since AB is a diameter, angle ACB = 90°. Using the angle sum of a triangle:
Angle ABC = 180° − 90° − 37° = 53°.
A complete answer says: “Angle ACB is 90° because it is the angle in a semicircle.” The diameter is essential: the same rule does not apply to every chord.
2. Tangent and radius
A tangent touches a circle at one point. The radius to that point is perpendicular to the tangent.
Example: PA is tangent to the circle at A, O is the centre, and angle OPA = 34°. Find angle AOP.
Angle OAP = 90° because OA meets tangent PA at A. Therefore:
Angle AOP = 180° − 90° − 34° = 56°.
Notice that two pieces of reasoning are needed: the tangent theorem and the angle sum of a triangle.
3. Angle at the centre and circumference
For angles standing on the same arc, the angle at the centre is twice the angle at the circumference. This is an Extended theorem.
Example: Central angle AOB = 116°. Point C is on the major arc AB, so angle ACB stands on the minor arc AB. Find angle ACB.
Angle ACB = 116° ÷ 2 = 58°.
Conversely, a circumference angle of 41° corresponds to a central angle of 82° standing on the same arc. Always identify the arc first. Depending on the location of the points, blindly halving an angle can give the wrong result.
4. Angles in the same segment
Two circumference angles standing on the same chord, with vertices on the same side of the chord, are equal.
Example: Points C and D lie on the same arc AB. If angle ACB = 42°, then angle ADB = 42°.
Mark the shared chord AB in your diagram. This theorem is especially useful when the given and unknown angles appear in different triangles.
5. Opposite angles of a cyclic quadrilateral
A cyclic quadrilateral has all four vertices on the circle. Its opposite angles add to 180°.
Example: ABCD is cyclic and angle DAB = 108°. Find angle BCD.
Angle BCD = 180° − 108° = 72°.
Do not confuse opposite angles with adjacent angles. First verify that all four vertices lie on the circumference.
6. Alternate segment theorem
The angle between a tangent and a chord equals the circumference angle in the opposite segment standing on that chord. For example, if the appropriate tangent–chord angle is 47°, the corresponding angle in the opposite segment is 47°.
A diagram has two possible tangent rays at the point of contact, producing supplementary angles. Label the actual angle carefully before applying the theorem.
Three original practice questions
Q1: AB is a diameter and angle CAB = 29°. Find angle ABC.
Answer: 180° − 90° − 29° = 61°.
Q2: A central angle on minor arc AB is 134°. C lies on the major arc AB. Find angle ACB.
Answer: 134° ÷ 2 = 67°.
Q3: ABCD is cyclic and angle ABC = 119°. Find angle ADC.
Answer: 180° − 119° = 61°.
How to revise effectively
For each question, follow four steps: identify the relevant feature, name the theorem, calculate, and write the reason. Parents can help by showing an unlabeled diagram and asking, “Which detail tells you what rule to use?” Students should be able to explain why the line is a diameter or tangent, or why the four points form a cyclic quadrilateral.
This approach builds transferable reasoning instead of relying on memorising a familiar picture. A short diagnostic can help identify whether the main difficulty is geometric vocabulary, recognising the theorem or calculating the angle.
Official source: Cambridge IGCSE Mathematics 0580 syllabus for 2025–2027, sections C4.7 and E4.7–E4.8.
