The Alternate Segment Theorem (Edexcel IGCSE Maths)

Alternate Segment Theorem

Spec point — 4.6C

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

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

Exam Hint 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.

$DAC$ is a straight line.

$EBF$ is a tangent to the circle.

Find the value of $x$.

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

Angle $x$and angle CAB form a straight lie

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

$x=133$