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Chapter 12: Geometry of straight lines

12.1 Introduction

In this chapter, we will explore the relationships between pairs of angles that are created when:

  • straight lines intersect (meet or cross)
  • lines are perpendicular to each other
  • a third line crosses two parallel lines
Imagine how many of the properties of angles and intersecting lines the architect had to use to be able to design this staircase.

You will come to understand what is meant by vertically opposite angles, corresponding angles, alternate angles and co-interior angles. You will then use your knowledge to help you work out unknown angles in geometric figures.


An angle is formed when two straight lines meet at a point. The point where the lines meet is called the vertex. Angles are labelled with a caret or 'hat' on a letter, for example, \(\hat{A}\).

the space between two straight lines that meet at a point
the point at which two straight lines meet

In this diagram, angle \(\hat{A}\) is formed between line segments \(BA\) and \(CA\). We can also name the angle according to the line segments that make up the angle, for example \(C\hat{A}B\) or \(B\hat{A}C\).

The \(\angle\) symbol is a short method of writing the word angle in geometry, and is often used in phrases such as "the sum of \(\angle\)s in \(\triangle\)".