Math Core

Railroad tracks, the lines on notebook paper and the edges of a ladder are all parallel. When another line crosses a pair of parallel lines, it creates eight angles, and the angles come in matching groups. Once you know one angle, you can find all the others.

Angle pairs you already know

When two lines cross, they form two pairs of vertical angles, the angles directly across from each other. Vertical angles are always equal.

Two angles that sit side by side and form a straight line are a linear pair. They are supplementary: their measures add to $180^\circ$ .

Transversals

A transversal is a line that crosses two or more other lines. In the diagram, line $t$ is a transversal crossing lines $m$ and $n$ , and $m \parallel n$ (read " $m$ is parallel to $n$ "). The eight angles are numbered.

Parallel lines m and n cut by transversal t.

The region between lines $m$ and $n$ is the interior; the regions outside them are the exterior. The angle pairs have names:

pair	what it means	in the diagram
corresponding angles	same position at each intersection	$\angle 1$ and $\angle 5$ , $\angle 2$ and $\angle 6$ , $\angle 3$ and $\angle 7$ , $\angle 4$ and $\angle 8$
alternate interior angles	between the lines, on opposite sides of $t$	$\angle 3$ and $\angle 6$ , $\angle 4$ and $\angle 5$
alternate exterior angles	outside the lines, on opposite sides of $t$	$\angle 1$ and $\angle 8$ , $\angle 2$ and $\angle 7$
same-side interior angles	between the lines, on the same side of $t$	$\angle 3$ and $\angle 5$ , $\angle 4$ and $\angle 6$

Why the angles match

Imagine translating line $m$ down along the transversal until it lands on line $n$ . Because the lines are parallel, $m$ fits exactly onto $n$ , and the whole top intersection lands on the bottom one. A translation doesn't change angle measures, so $\angle 1$ lands on $\angle 5$ , $\angle 2$ on $\angle 6$ , and so on. That's why corresponding angles are equal. The other rules follow from that one plus vertical angles and linear pairs.

Parallel lines cut by a transversal

If two parallel lines are cut by a transversal, then:

corresponding angles are equal,
alternate interior angles are equal,
alternate exterior angles are equal,
same-side interior angles are supplementary (they add to $180^\circ$ ).

In practice, this means every angle in the picture is one of just two sizes. All the "small" angles are equal, all the "large" angles are equal, and a small one plus a large one is $180^\circ$ .

Worked example: Finding all eight angles

In the diagram, $m\angle 2 = 56^\circ$ . Find the other seven angles.

$\angle 3$ is vertical to $\angle 2$ , so $m\angle 3 = 56^\circ$ .
$\angle 1$ and $\angle 2$ form a linear pair, so $m\angle 1 = 180^\circ - 56^\circ = 124^\circ$ . Its vertical angle $\angle 4$ is also $124^\circ$ .
The bottom intersection is a copy of the top one: $m\angle 6 = m\angle 7 = 56^\circ$ and $m\angle 5 = m\angle 8 = 124^\circ$ .

Worked example: Alternate interior angles

Two parallel lines are cut by a transversal. A pair of alternate interior angles measure $(3x + 10)^\circ$ and $(5x - 30)^\circ$ . Find $x$ and the angle measure.

Alternate interior angles are equal, so set the expressions equal:

\begin{aligned} 3x + 10 &= 5x - 30 \\ 40 &= 2x \\ x &= 20 \end{aligned}

Each angle measures $3(20) + 10 = 70^\circ$ . Check: $5(20) - 30 = 70$ . ✓

Worked example: Same-side interior angles

Two parallel lines are cut by a transversal. A pair of same-side interior angles measure $(2x + 15)^\circ$ and $(4x - 3)^\circ$ . Find both angles.

Same-side interior angles are supplementary:

\begin{aligned} (2x + 15) + (4x - 3) &= 180 \\ 6x + 12 &= 180 \\ x &= 28 \end{aligned}

The angles are $2(28) + 15 = 71^\circ$ and $4(28) - 3 = 109^\circ$ . Check: $71 + 109 = 180$ . ✓

Common mistake

Same-side interior angles are supplementary, not equal. Before you write an equation, decide whether the two angles look the same size (set them equal) or one looks acute and one obtuse (make them add to $180$ ). Also, these rules only work when the lines are parallel.

Practice

The problems that refer to the diagram use the diagram in the Transversals section, where $m \parallel n$ .

Practice 1

In the diagram, what kind of angle pair are $\angle 1$ and $\angle 5$ ?

Practice 2

In the diagram, $m\angle 5 = 118^\circ$ . What is $m\angle 4$ , in degrees?