What shape must you look for?

A **cyclic triangle** — all three vertices on the circumference — where **one vertex meets a tangent**.

Where is the equal angle located?

**Inside the triangle** at the corner **opposite** the side that forms the angle with the tangent.

What exam phrase can you use?

**The Alternate segment theorem** — or the full statement about the angle between chord and tangent equalling the angle in the alternate segment.

In the first worked example, angle CBF = angle CAB. Why?

**Alternate segment theorem** — the angle between the tangent (BF) and the chord (BC) equals the angle in the alternate segment (CAB).

What equation gives x = 133 in the first example?

**x + 47 = 180** — angle x and angle CAB form a straight line, so they are supplementary.

What other theorem is often combined with the alternate segment?

The **angle at the centre is twice the angle at the circumference** — when both appear in the same diagram.

How frequently does this theorem appear in exams?

**Frequently** — examiner tip: questions involving the alternate segment theorem often appear in exams.

Which vertex of the cyclic triangle must touch the tangent?

**At least one** — the theorem applies when one vertex of the cyclic triangle meets the tangent.

The Alternate Segment Theorem | Pearson Edexcel IGCSE Maths

Q: What is the alternate segment theorem?

The **angle between a chord and a tangent** is equal to the **angle in the alternate segment** (the opposite side of the chord).

Q: What is the alternate segment?

The region on the **opposite side** of the chord from the angle formed between the chord and the tangent.

Geometry & Trigonometry

The Alternate Segment Theorem

Summary

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 the chord from the angle formed between the chord and the tangent. Look for a cyclic triangle (all vertices on the circumference) where one vertex meets a tangent. The angle between the tangent and the chord equals the angle inside the triangle at the opposite vertex. This theorem frequently appears in exams.

1. The Theorem

Circle 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 the chord from the angle formed between the chord and the tangent.

Angle between tangent and chord BC = angle BAC (in alternate segment).

2. How to Spot

Look for a cyclic triangle — all three vertices on the circumference.

One vertex of the triangle must meet a tangent.

To identify equal angles: mark the angle between the tangent and the side of the cyclic triangle.

The equal angle is inside the triangle at the corner opposite the side that forms the first angle.

3. Worked Example (Type 1)

Problem: A, B, C on circle. DAC is a straight line. EBF is tangent. Find x.

Angle CBF = angle CAB (alternate segment theorem). So angle CAB = 47°.

Angle x and angle CAB form a straight line: x + 47 = 180 → x = 133°.

4. Worked Example (Type 2)

Problem: Find x. Tangent at R. Angle between QR and tangent = 2x + 5.

By alternate segment theorem: angle RQP = 2x + 5.

Angle RPQ and angle ROQ both from same chord: angle ROQ = 2(2x + 5). Angle at centre is twice angle at circumference.

2(2x + 5) = 5x − 2 → 4x + 10 = 5x − 2 → x = 12.

The Alternate Segment Theorem

Summary

1. The Theorem

2. How to Spot

3. Worked Example (Type 1)

4. Worked Example (Type 2)

5. Examiner Tips

The Alternate Segment Theorem

Summary

1. The Theorem

2. How to Spot

3. Worked Example (Type 1)

4. Worked Example (Type 2)

5. Examiner Tips