Parallel lines

​​In a nutshell

Parallel lines are lines that will never meet. Angles in parallel lines have their own set of rules you need to learn.

Definition

Two lines are parallel if they will never meet (or intersect). Parallel lines are lines that are moving in the exact same direction.

Representing parallel lines

A pair of parallel lines can be represented by matching arrowheads on both lines.

Vertically opposite angles

Two straight lines intersecting create two pairs of vertically opposite angles. What you need to know is that vertically opposite angles are equal in size.

​$$\alpha=\gamma,\beta=\delta$$​​

Alternate, interior and corresponding angles

When two parallel lines are intersected by a third line, pairs of angles can appear in three different forms. Each of these forms have their own rules.

NAME	INFORMAL NAME	DIAGRAM	RULE
Alternate angles	Z-angles		Alternate angles are equal
Interior angles	C-angles		Interior angles add up to $$180\degree$$​​
Corresponding angles	F-angles		Corresponding angles are equal

Note: The informal names are given to make it easier to identify and remember the rules. When solving questions, you must use the actual names of each type.

Angle problems

To solve problems involving angles in parallel lines, you need to be able to identify when you can use a rule that helps you find out more angles in the question.

Example

In the diagram below, the lines $$m$$ and $$n$$ are parallel, as are the lines $$r$$ and $$s$$. If $$a=70\degree$$, find the values of $$e,c$$ and $$d$$.

Identify any angle pairs that form a rule.

Angles $$a$$ and $$e$$ are F-angles. So, they must be equal.

​$$e=a=70\degree$$​, as they are corresponding angles.

Angles $$e$$ and $$c$$ are Z-angles. So, they also must be equal.

$$c=e=70\degree$$​, as they are alternate angles.

Angles $$d$$ and $$c$$ are angles on a straight line. So, they must add up to $$180\degree$$.

​$$d=180-c=180-70=110\degree$$​, as they are angles on a straight line.

​$$\underline{e=70\degree,c=70\degree, d = 110\degree}$$​​