# Year 10+ Lines and Angles

### Chapters

### Parallel Lines and Pairs of Angles

# Parallel Lines and Pairs of Angles

Two lines are `parallel`

if they always stay the same distance apart and never cross each other.

In the examples below, the orange and blue lines are parallel

Example 1

Example 2

## Pairs of Angles and Parallel Lines

A line that crosses two or more other lines is called a `transversal`

. When a pair of parallel lines is crossed by a transversal, a number of different
pairs of angles are formed, many of which are equal, while others are supplementary (add to \(180^\circ\)). The diagrams below show these different angles.

In the picture on the left, the blue lines are parallel and the orange line is a transversal. In addition:

- The pairs of angles labelled by \(a\) and \(b\) are
`alternate angles`

- The pairs of angles labelled by \(x\) and \(y\) are
`alternate exterior angles`

In the picture on the right, the blue lines are parallel and the orange line is a transversal. In addition:

- The pairs of angles labelled by \(c\), \(d\), \(e\) and \(f\) are
`corresponding angles`

- The pairs of angles labelled by \(c\) and \(f\), and \(d\) and \(e\) are
`cointerior angles`

## Tests for Parallel Lines

Each type of angle pair gives rise to a test for parallel lines:

- If the two angles in any pair of alternate angles formed when a transversal cuts a pair of lines are
**equal**, then the pair of lines is**parallel** - If the two angles in any pair of corresponding angles formed when a transversal cuts a pair of lines are
**equal**, then the pair of lines is**parallel** - If the two angles in any pair of alternate exterior angles formed when a transversal cuts a pair of lines are
**equal**, then the pair of lines is**parallel** - If the two angles in any pair of cointerior angles formed when a transversal cuts a pair of lines are
**supplementary**, then the pair of lines is**parallel**

So, if you have two lines cut by a transversal and you find a pair of alternate angles that are **not** equal, or a pair of alternate exterior angles
that are **not**
equal, or a pair of corresponding lines that are **not** equal, or a pair of cointerior angles that are
**not** supplementary, then you know that the two lines are **not**
parallel.

Let's look at some examples.

### Examples

In the picture, both alternate angles marked in green are equal to \(60^\circ\). So, the two blue lines are parallel.

In the picture, one of the alternate angles is equal to \(124^\circ\), but the other is equal to \(122^\circ\). The two blue lines are
**not** parallel.

The picture shows a pair of corresponding angles that are equal (to \(60^\circ\)). Therefore, the two blue lines are parallel.

In the picture, one of the corresponding angles marked is \(124^\circ\), while the other is \(122^\circ\). As these
corresponding angles are unequal, the two blue lines are **not** parallel.

The picture shows a pair of cointerior angles that is supplementary (\(120^\circ + 60^\circ = 180^\circ\)). Therefore, the two blue lines are parallel.

In the picture, the pair of cointerior angles adds up to \(57^\circ + 124^\circ = 181^\circ \neq 180^\circ\). As this pair of
cointerior angles is **not** supplementary, the two blue lines are **not** parallel.

### Description

In this mini book, you will learn about

- Alternate angles
- Cointerior angles
- Radians
- Vertically Opposite Angles

and several other topics related to lines and angles.

### Audience

Year 10 or higher

### Learning Objectives

To learn the basics of Lines and Angles stream of Geometry

Author: Subject Coach

Added on: 28th Sep 2018

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