GCSE MATHS • HIGHER TIER
Alternate Segment Theorem Made Simple
Learn how to identify tangent-chord angles, match them with angles in the alternate segment, and solve GCSE circle theorem questions.
Clear method
Exam practice
Worked proofs
Alternate Segment Theorem in Four Moves
1
Identify
Identify the tangent and chord.
2
Locate
Find the angle in the opposite segment.
3
Match
Set the two angles equal.
4
Solve
Calculate the missing angle or solve the algebra.
What You’ll Learn
Understand the Theorem
Understand the link between a tangent-chord angle and the alternate segment angle.
Recognise the Diagram
Identify tangents, chords and the matching angle inside the circle.
Apply the Key Rule
Use equal angles to solve missing-angle questions.
Solve Algebraic Questions
Form and solve equations involving equal angles.
Connect Circle Theorems
Combine this theorem with other circle theorem rules.
Key Rules
Angle between tangent and chord = angle in the alternate segment
Understanding the Theorem
Why Is This Theorem Important?
Worked Proof: Alternate Segment Theorem
Step-by-Step Method
- 1. Identify the tangent.
- 2. Find the chord connected to the tangent.
- 3. Locate the angle in the opposite segment.
- 4. Apply the theorem.
- 5. Set the angles equal.
- 6. Solve any calculations or algebra.
More Worked Examples
1
Question: The angle between a tangent and a chord is 62°.
The angle in the alternate segment is equal.
2
Question: The angle in the alternate segment is 48°.
Therefore, the tangent-chord angle is also 48°.
3
Question: The angle in the alternate segment is 95°.
The tangent-chord angle is equal.
4
Question: The tangent-chord angle is x and the alternate segment angle is 70°.
x = 70
- Common GCSE Mistakes
- Wrong theorem: Do not confuse it with angles in the same segment.
- Missing tangent: Check that a tangent is present.
- Wrong angle: Match the correct opposite angle.
- Algebra errors: Set equal angles correctly.
- Diagram errors: Read marked angles carefully.
- Exam Tips
- Spot the tangent first: It is the main clue.
- State the theorem: Use its name in reasoning questions.
- Mark equal angles: Show the matching pair clearly.
- Check the segment: Use the opposite angle.
- Show algebra clearly: Write the equation before solving.
Topics to Revise Next
Tangent Theorems
Angles at the Centre
Angles in the Same Segment
Cyclic Quadrilaterals
Radius and Tangent Theorem
Circle Geometry
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Alternate Segment Theorem Tutorial
Step-by-step walkthroughs of Alternate Segment Theorem with clear methods and exam tips.
Frequently Asked Questions
The angle between a tangent and a chord is equal to the angle in the alternate segment.
A line that touches a circle at exactly one point.
A straight line joining two points on the circumference.
Look for a tangent, a chord meeting it and another angle in the opposite segment.
