Parallel lines and a transversal

Draw two par­al­lel lines, like the rails of a track, and then draw a third line that crosses both of them, like a road going over the rails. Eight angles appear. When the first two lines are par­al­lel, these eight angles are tied together in a very neat way: if you know just one of them, you can find all the oth­ers. This les­son shows you how.

The trans­ver­sal#

A line that crosses two or more lines at dif­fer­ent points is called a trans­ver­sal.

Let the two lines be $l$ and $m$, and let the trans­ver­sal be $t$. Pic­ture $l$ drawn across the top of your page and $m$ across the bot­tom, with $t$ slant­ing down through both.

At the point where $t$ crosses $l$ there are four angles. Name them like this:

• $\angle 1$: above $l$, to the left of $t$
• $\angle 2$: above $l$, to the right of $t$
• $\angle 3$: below $l$, to the left of $t$
• $\angle 4$: below $l$, to the right of $t$

Where $t$ crosses $m$ there are four more, named the same way:

• $\angle 5$: above $m$, left of $t$; $\angle 6$: above $m$, right of $t$
• $\angle 7$: below $m$, left of $t$; $\angle 8$: below $m$, right of $t$

Inte­rior angles lie between the two lines: $\angle 3$, $\angle 4$, $\angle 5$ and $\angle 6$.

Exte­rior angles lie out­side the two lines: $\angle 1$, $\angle 2$, $\angle 7$ and $\angle 8$.

The pairs of angles#

Cor­re­spond­ing angles are in the same posi­tion at the two cross­ings:

$$\angle 1 \text{ and } \angle 5, \quad \angle 2 \text{ and } \angle 6, \quad \angle 3 \text{ and } \angle 7, \quad \angle 4 \text{ and } \angle 8$$

Alter­nate inte­rior angles are between the lines, on oppo­site sides of the trans­ver­sal:

$$\angle 3 \text{ and } \angle 6, \qquad \angle 4 \text{ and } \angle 5$$

Alter­nate exte­rior angles are out­side the lines, on oppo­site sides of the trans­ver­sal:

$$\angle 1 \text{ and } \angle 8, \qquad \angle 2 \text{ and } \angle 7$$

Co-inte­rior angles (also called inte­rior angles on the same side) are between the lines, on the same side of the trans­ver­sal:

$$\angle 3 \text{ and } \angle 5, \qquad \angle 4 \text{ and } \angle 6$$

When the lines are par­al­lel#

If $l \parallel m$, then:

1. Cor­re­spond­ing angles are equal.
2. Alter­nate inte­rior angles are equal.
3. Alter­nate exte­rior angles are equal.
4. Co-inte­rior angles add up to $180^\circ$.

Here is why rule 2 fol­lows from rule 1. $\angle 3$ and $\angle 2$ are ver­ti­cally oppo­site, so $\angle 3 = \angle 2$. And $\angle 2 = \angle 6$, because they are cor­re­spond­ing angles. So $\angle 3 = \angle 6$.

And rule 4: $\angle 3 = \angle 7$ (cor­re­spond­ing), and $\angle 5 + \angle 7 = 180^\circ$ (a lin­ear pair along $m$). So $\angle 5 + \angle 3 = 180^\circ$.

Test­ing whether two lines are par­al­lel#

The rules also work the other way round. A trans­ver­sal crosses two lines. If

• a pair of cor­re­spond­ing angles is equal, or
• a pair of alter­nate inte­rior angles is equal, or
• a pair of co-inte­rior angles adds up to $180^\circ$,

then the two lines are par­al­lel. If none of these is true, the lines are not par­al­lel, and if they are drawn long enough, they will meet.

Exam­ple 1: Find­ing all eight angles#

$l \parallel m$ and $\angle 1 = 65^\circ$. Find the other seven angles.

At the first cross­ing:

• $\angle 4 = 65^\circ$, ver­ti­cally oppo­site to $\angle 1$.
• $\angle 2 = 180^\circ - 65^\circ = 115^\circ$, a lin­ear pair with $\angle 1$.
• $\angle 3 = 115^\circ$, ver­ti­cally oppo­site to $\angle 2$.

At the sec­ond cross­ing, each angle equals its cor­re­spond­ing angle:

• $\angle 5 = \angle 1 = 65^\circ$ and $\angle 6 = \angle 2 = 115^\circ$.
• $\angle 7 = \angle 3 = 115^\circ$ and $\angle 8 = \angle 4 = 65^\circ$.

Check with the other rules. Alter­nate inte­rior: $\angle 4 = \angle 5 = 65^\circ$. Co-inte­rior: $\angle 3 + \angle 5 = 115^\circ + 65^\circ = 180^\circ$. All cor­rect.

Notice: with par­al­lel lines, every one of the eight angles is either $65^\circ$ or $115^\circ$. The two sizes always add up to $180^\circ$.

Exam­ple 2: Co-inte­rior angles#

$l \parallel m$. Two co-inte­rior angles are $(2x + 10)^\circ$ and $(3x - 5)^\circ$. Find $x$ and both angles.

Co-inte­rior angles add up to $180^\circ$:

$$\begin{aligned} (2x + 10) + (3x - 5) &= 180 \\ 5x + 5 &= 180 \\ 5x &= 175 \\ x &= 35 \end{aligned}$$

The angles are $2 \times 35 + 10 = 80$ degrees and $3 \times 35 - 5 = 100$ degrees.

Exam­ple 3: Alter­nate angles#

$l \parallel m$. Two alter­nate inte­rior angles are $(4x - 20)^\circ$ and $(2x + 30)^\circ$. Find $x$ and the angle.

Alter­nate inte­rior angles are equal:

$$\begin{aligned} 4x - 20 &= 2x + 30 \\ 2x &= 50 \\ x &= 25 \end{aligned}$$

Each angle is $4 \times 25 - 20 = 80$ degrees.

Exam­ple 4: Are the lines par­al­lel?#

A trans­ver­sal crosses lines $p$ and $q$.

(a) A pair of alter­nate inte­rior angles are $72^\circ$ and $72^\circ$. Are $p$ and $q$ par­al­lel?

(b) A pair of co-inte­rior angles are $75^\circ$ and $95^\circ$. Are $p$ and $q$ par­al­lel?

(a) Yes. The alter­nate inte­rior angles are equal, so $p \parallel q$.

(b) No. $75^\circ + 95^\circ = 170^\circ$, not $180^\circ$. So $p$ and $q$ are not par­al­lel.

Prac­tice#

1. Name the pair of angles: $\angle 2$ and $\angle 6$.
2. Name the pair of angles: $\angle 4$ and $\angle 5$.
3. Name the pair of angles: $\angle 4$ and $\angle 6$.
4. $l \parallel m$ and $\angle 2 = 120^\circ$. Find $\angle 6$, $\angle 7$ and $\angle 5$.
5. $l \parallel m$ and $\angle 3 = 48^\circ$. Find all the other angles.
6. $l \parallel m$. Two cor­re­spond­ing angles are $(3x + 15)^\circ$ and $(5x - 25)^\circ$. Find $x$ and the angle.
7. $l \parallel m$. Two co-inte­rior angles are $(x + 40)^\circ$ and $(2x + 20)^\circ$. Find both angles.
8. A trans­ver­sal crosses two lines. A pair of cor­re­spond­ing angles are $58^\circ$ and $62^\circ$. Are the lines par­al­lel?
9. A trans­ver­sal crosses two lines. A pair of co-inte­rior angles are $108^\circ$ and $72^\circ$. Are the lines par­al­lel?

Answers#