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.​