4.6 Angles

Cambridge IGCSE Mathematics 0580

2026 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Calculate unknown angles using basic angle properties: angles at a point (360°), angles on a straight line (180°), vertically opposite angles, angle sum of a triangle (180°), and angle sum of a quadrilateral (360°).
  2. Calculate unknown angles in parallel lines: corresponding angles (equal), alternate angles (equal), and co-interior angles (add up to 180°).
  3. Know and use angle properties of regular polygons, including interior and exterior angles and their sums.
  4. Know and use angle properties of both regular and irregular polygons.
  5. Use three-letter notation for angles (e.g. angle ABC) and always give correct geometric reasons for your answers.

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Part 1 — Basic Angle Properties

1.1 Three-Letter Angle Notation

In geometry, we often label angles using three letters, like angle ABC. Here is how it works:

  • The middle letter is always the vertex — that is, the point where the angle is actually located (the corner).
  • The first and last letters are points on the two lines that form the angle.

So if you have a point B with lines going to point A and point C, the angle at B is written as angle ABC (or sometimes ∠ABC).

Example: If three points are labelled A, B, and C, and you want to talk about the angle at the corner B, you write it as angle ABC. The angle is "sitting" at B.

This notation is important — in the exam, you must use it correctly when referring to specific angles in diagrams.

1.2 Angles at a Point

When several angles meet at a single point and together they form a complete turn (going all the way around), they add up to 360°.

Rule: The sum of angles at a point = 360°

Think of it like slicing a pizza — no matter how many slices you cut, all the slice angles together always make a full circle of 360°.

Example: Three angles meet at a point. Two of them are 120° and 95°. Find the third angle.

120° + 95° + x = 360° 215° + x = 360° x = 145°

Reason you must write: "Angles at a point sum to 360°"

1.3 Angles on a Straight Line

When angles are formed on one side of a straight line and they all sit on that line at the same point, they add up to 180°.

Rule: The sum of angles at a point on a straight line = 180°

Imagine a straight road — if you draw lines shooting upward from one spot on the road, all those angles together only cover half of a full turn, which is 180°.

Example: Two angles are on a straight line. One is 65°. Find the other.

65° + x = 180° x = 115°

Reason: "Angles on a straight line sum to 180°"

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