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O Level · Mathematics 0580

Angles in Polygons & Parallel Lines

Revise Angles in Polygons & Parallel Lines for Mathematics 0580 (O Level) — revision notes, 351 practice questions and instant AI marking.

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The Big Idea

Angles follow predictable patterns based on the shapes they're in and the way lines interact. If you know these patterns, you can always find any missing angle.

What You'll Learn

Basic angle rules: How angles work around points, on lines, and in triangles
Polygon angles: A formula to find the sum of angles in any polygon
Regular polygons: How to find each angle when all sides are equal
Parallel lines: Three angle patterns that emerge when a line cuts through parallel lines
Solving problems: Using all of this to find missing angles in complex diagrams

1. Basic Angle Properties

How to Label Angles and Shapes Correctly

Before you can solve angle problems, you need to label things properly. This sounds simple, but it matters for communication and avoiding mistakes.

Line segment AB: The straight line going from point A to point B
Angle ABC: The angle at point B, formed by the rays BA and BC. Always write the vertex (the corner) in the middle
Triangle ABC: Label the corners by following the sides, either clockwise or anticlockwise. Both ABCD and ADCB are valid for a quadrilateral — but ABDC is not, because it jumps around the shape

Why does this matter? When you label angle ABC, putting B in the middle tells anyone reading your work exactly which corner you mean. This is crucial when a diagram has many angles.

Angles Around a Point Add Up to 360°

Imagine standing at a point and spinning around. You rotate all the way around once. That full rotation is 360 degrees. If multiple angles meet at a point, they must add up to that full turn.

Rule

All angles around a point sum to 360°

Example

Three angles meet at a point. Two of them are 110° and 85°. What is the third?

Third angle = 360° − 110° − 85° = 165°

Angles on a Straight Line Add Up to 180°

A straight line is actually a 180° turn. If you have angles sitting on a straight line, side by side, they share that 180°. This is sometimes called a "linear pair" or "supplementary angles".

Rule

All angles on a straight line sum to 180°

Example

Two angles on a straight line are 67° and x°. Find x.

67° + x° = 180°

x° = 180° − 67° = 113°

Quick check: If you see angles sitting on what looks like a straight line in a diagram, and they don't obviously look like they'd add to 180°, re-examine the diagram. The line might bend, or one of the angles might be outside the pair. Look carefully.

Vertically Opposite Angles Are Equal

When two straight lines cross, they form four angles. The two angles that are across from each other (not next to each other) are called vertically opposite. These are always equal.

Why? Look at it this way: if you take two opposite angles, each one sits on a straight line with the same two neighbours. Since both pairs must add to 180°, the opposite angles must be equal.

Rule

Vertically opposite angles are equal

Two lines crossing: \ / \ a / \ / d X b / \ / \ / c \ Angles a and c are vertically opposite → a = c Angles b and d are vertically opposite → b = d Also: a + b = 180° (on a straight line)

Example

Angles in a Triangle

2. Angles in Polygons

What Is a Polygon?

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Also in the full note

3. Angles in Parallel Lines
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
Final Challenge Questions
Angles in a Quadrilateral
Practice Questions
Interior and Exterior Angles

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0580 Nov 2025 · Paper 1 · Variant 2 Cambridge · mark scheme Open → 0580 Nov 2025 · Paper 1 · Variant 3 Cambridge · mark scheme Open → 0580 Nov 2025 · Paper 2 · Variant 1 Cambridge · mark scheme Open → 0580 Nov 2025 · Paper 3 · Variant 1 Cambridge · mark scheme Open → 0580 Nov 2025 · Paper 4 · Variant 3 Cambridge · mark scheme Open → 0580 Mar 2025 · Paper 3 · Variant 2 Cambridge · mark scheme Open →

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Angles in Polygons & Parallel Lines