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.

Practice lesson for the WJEC Mathematics and Numeracy Double Award. This is not an official WJEC page.

Double Award guide·WJEC / CBAC

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°

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