Circle Theorems: Angles in the Alternate Segment

What Is the Alternate Segment Theorem?

The alternate segment theorem links tangents and angles inside a circle.

It says:

The angle between a tangent and a chord is equal to the angle in the alternate segment of the circle.

This only works when you have a triangle inside the circle and a tangent touching the circle at one of the triangle's vertices.

How to Spot It

• Look for a triangle where all three corners lie on the circumference (a cyclic triangle)
• One side of the triangle will be a chord
• A tangent touches the circle at one of the triangle’s points
• The angle between the tangent and the chord is equal to the angle opposite inside the triangle

Example

Question:

In the diagram below:

A triangle $PQR$ is inscribed in a circle

A tangent touches the circle at point $R$

The angle between the tangent and side $QR$ is $43\degree$

Find $\angle QPR$.

 
Answer:

By the alternate segment theorem:

$\angle QPR = 43\degree$

Reason:

The angle between a tangent and a chord equals the angle in the alternate segment.

Worked Example

Question:

In a circle with centre $O$, points $P, Q, R$ lie on the circumference. A tangent touches the circle at point $R$. You are told:

$\angle QOR = 5x - 2$

The angle between the tangent and chord $QR$ is $2x + 5$

Find the value of $x$.

Step 1: Apply the Alternate Segment Theorem

The angle between the tangent and chord $QR$ is $2x + 5$. So:

$\angle RPQ = 2x + 5$

 
Step 2: Use the Angle at Centre Theorem

$\angle QOR$ is at the centre, and it's made by the same arc as $\angle RPQ$.

So:

$\angle QOR = 2 \times \angle RPQ$

Substitute the expressions:

$5x - 2 = 2(2x + 5)$

Step 3: Solve the Equation

Expand the right-hand side:

$5x - 2 = 4x + 10$

Subtract $4x$ from both sides:

$x - 2 = 10$

Add 2 to both sides:

$x = 12$

Final Answer:

$\boxed{x = 12}$