Circle Theorems (Copy)

Cheat Sheet: Angle Properties of Circles (IGCSE Mathematics 0580 – CORE)
Topic: Angles in Semicircles and Tangents

1. Angle in a Semicircle

• Rule: The angle formed in a semicircle is always 90°.
• If you draw a triangle where one side is the diameter of the circle, the angle opposite the diameter (at the circle’s edge) is always a right angle.

Key Points:

• The triangle is a right-angled triangle.
• The right angle is at the point on the circle, not at the centre.

Example:
If a triangle is drawn with the diameter as one side, and the third vertex lies anywhere on the circumference, the angle at that vertex is 90°.

2. Angle Between Tangent and Radius

• Rule: The angle between a tangent and the radius drawn to the point of contact is always 90°.

Key Points:

• A tangent is a straight line that touches the circle at exactly one point.
• The radius drawn to the point where the tangent touches the circle is perpendicular to the tangent.

Example:
If you are given a circle with a tangent touching it at point P, and the radius OP is drawn, then angle O-P-Tangent = 90°.

3. Important Notes

• Always check whether a side given is a diameter or a radius.
• Semicircle → 90° inside triangle.
• Tangent and Radius → 90° at point of contact.
• Diagrams must show perpendicular (right angle) marks where necessary.

4. Quick Reference

This covers all CORE-level circle angle properties you need for calculating unknown angles and giving correct explanations.