# Angles in parallel lines

## In a nutshell

Angle geometry involves finding the link between different angles when a line called a 'traversal' intersects a pair of parallel lines. Vertically opposite angles occur when two lines intersect forming a cross. When a traversal intersects a pair of parallel lines, it is possible to identify corresponding angles, alternate angles and interier angles. These angle rules can then be used to help find missing angles.

## Angle rules

There are four angle rules: vertically opposite angles, corresponding angles, alternate angles and interior angles. Some of these rules have informal names, such as $F$ angles or $Z$ angles, as these help to identify and remember the rules. However, when answering questions, make sure to use the actual names to describe the rule.

**NAME** | **DESCRIPTION** | **DIAGRAM** |

Vertically opposite angles | Vertically opposite angles are equal to each other. $\angle x = \angle y$ | |

Corresponding angles ($F$ angles) | Corresponding angles are equal to each other. $\angle x = \angle y$ | |

Alternate angles ($Z$ angles) | Alternate angles are equal to each other. $\angle x = \angle y$ | |

Interior angles ($C$ angles) | Interior angles add up to $180\degree$. $\angle x + \angle y = 180\degree$ | |

## Angle problems

The rules above can be used to identify missing angles.

##### Example 1

*The diagram shows the angles formed around the parallel lines $s$ and $r$, with traversal $t$. Given that $\angle a= 120\degree$, find the size of angles $c, e$ and $f$.*

*$\angle c$ is vertically opposite $\angle a$, so*

$\angle c = \angle a = \underline{120 \degree}$

$\angle e$ *and* $\angle a$ *are corresponding, so *

$\angle e = \angle a = \underline{120 \degree}$

$\angle f$ *and* $\angle c$ *are interior angles, so *

$\begin{aligned}\angle f + \angle c &= 180\degree \\\angle f &= 180\degree - \angle c \\\angle f &= 180\degree - 120\degree \\\angle f &= \underline{60\degree}\end{aligned}$

**Note: **$\angle f$ could have also been found using the fact that angles on a straight line add to $180\degree$, as $\angle e$ had already been found.

##### Example 2

*The diagram shows the angles formed from two sets of parallel lines. The line $m$ is parallel to $n$, and $r$ is parallel to $s$. State which angles are equal to $\angle a$ and give reasons.*

$\underline{\angle e = \angle a}$ *as corresponding angles are equal. *

* $\underline{\angle c = \angle e \ (= \angle a)}$ as alternate angles are equal.*

*$\underline{\angle b = \angle c \ (= \angle a)}$ as vertically opposite angles are equal.*