Circle theorems describe relationships between angles, chords, tangents, and arcs. These are essential Higher GCSE topics.
The Theorems
1. Angle at the Centre
The angle at the centre is twice the angle at the circumference (same arc).
2. Angle in a Semicircle
An angle inscribed in a semicircle is 90°.
3. Angles in the Same Segment
Angles subtended by the same arc at the circumference are equal.
4. Cyclic Quadrilateral
Opposite angles of a cyclic quadrilateral sum to 180°.
5. Tangent-Radius
A tangent is perpendicular to the radius at the point of contact.
6. Two Tangents from a Point
Tangent segments from an external point are equal in length.
7. Alternate Segment Theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
Strategy
- Identify the relevant theorem.
- State which theorem you're using (examiners require this).
- Calculate the angle.
Worked Example: Example 1
Angle at centre = . Angle at circumference = (Theorem 1).
Worked Example: Example 2
Cyclic quadrilateral: opposite angle to is (Theorem 4).
Worked Example: Example 3
Tangent from , , radius = 5. Distance (Theorem 5 + Pythagoras).
Practice Problems
- Angle at centre is . Find the angle at the circumference.
- In a cyclic quad, three angles are , , . Find the fourth.
- Prove that the angle in a semicircle is .
Want to check your answers and get step-by-step solutions?
Key Takeaways
Centre angle = 2 × circumference angle.
Semicircle → 90°.
Same segment → equal angles.
Cyclic quad → opposite angles sum to 180°.
Tangent ⊥ radius.
Always state the theorem in your working.