Unit 2 – Geometry (Higher) · Higher
Circle Theorems
The main circle theorems tested at Higher tier: angles in the same segment, centre and circumference, cyclic quadrilaterals, tangent and radius, alternate segment.
Circle theorems give fixed relationships between angles and lines in a circle. Learn the statements and be ready to give reasons in exam answers.
Key theorems
1. **Angle in a semicircle** is a right angle (angle in a semicircle = 90°).
2. **Angle at the centre** is twice the angle at the circumference (same arc).
3. **Angles in the same segment** are equal.
4. **Opposite angles in a cyclic quadrilateral** sum to 180°.
5. **Tangent is perpendicular to the radius** at the point of contact.
6. **Two tangents from an external point** are equal in length.
7. **Alternate segment theorem** – angle between tangent and chord equals angle in the alternate segment.
Worked example
A, B and C lie on a circle with diameter AC. Angle ABC is…?
Solution
**90°** – angle in a semicircle is a right angle.
Exam technique
Always state the theorem you are using as a reason.
Diagrams are not drawn to scale – do not assume sizes from the picture.
Mark equal angles and radii clearly when working.
Practice questions
Try these without looking at the answers first.
Q1.Angle at the centre is 110°. What is the angle at the circumference on the same arc?
Show answer
55°
Q2.In a cyclic quadrilateral one angle is 70°. Opposite angle?
Show answer
110°
Q3.A tangent meets a radius at the point of contact. What is the angle between them?
Show answer
90°