GCSEMathsWJEC EduqasHigherRevision Notes4. Geometry & MeasuresCircle TheoremsThe Alternate Segment Theorem

The Alternate Segment Theorem (WJEC Eduqas GCSE Maths: Higher): Revision Note

Exam code: C300

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 23 January 2026

Alternate segment theorem

Circle theorem: The alternate segment theorem

The angle between a chord and a tangent is equal to the angle in the alternate segment
- The alternate segment is the region on the opposite side of a chord from a given angle formed between that chord and a tangent line

Alternate Segment Theorem, IGCSE & GCSE Maths revision notes — **Example of angles in alternate segments**

To spot this circle theorem on a diagram
- look for a cyclic triangle
  - where all three vertices of the triangle lie on the circumference
- one vertex of the triangle meets a tangent
To identify which angles are equal
- mark the angle between the tangent and the side of the cyclic triangle
- the angle inside the triangle at the corner opposite the side of the triangle that forms the first angle is the equal angle
When explaining this theorem in an exam you can just say the phrase:
- The Alternate segment theorem

Examiner Tips and Tricks

Look for cyclic triangles and tangents in busy diagrams.

Questions involving the alternate segment theorem frequently appear in exams!

Worked Example

$A$ , $B$ and $C$ are points on a circle.

$D A C$ is a straight line.

$E B F$ is a tangent to the circle.

Find the value of $x$ .

Circle with triangle ABC inside, angles marked x° at A and 47° at B. Lines DE and CF are extended tangents. Text reads "Not drawn to scale."

Answer:

One vertex of this triangle meets a tangent at point B
The angle between one of its sides (BC) and the tangent is given
Find the angle inside the triangle, opposite to the same side (BC)

Angle CBF = Angle CAB by the alternate segment theorem

Circle with points A, B, C on circumference, lines AD, CF, AE, BF tangent. Angles at A and B are 47 degrees. Note: Not drawn to scale.

Angle $x$ and angle CAB form a straight lie

$\begin{array}{rcl} x + 47 & = & 180 \\ x & = & 180 - 47 \end{array}$

$x = 133$

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