Math Core

Angles rarely appear alone. When lines cross or rays share a vertex, angles come in pairs, and certain pairs always have a fixed relationship. If you know one angle in the pair, you know the other, which is the key to finding missing angles in almost every diagram.

Adjacent angles

Two angles are adjacent if they share a vertex and a side but have no interior points in common. In other words, they sit side by side without overlapping. In the figure below, $\angle ABD$ and $\angle DBC$ are adjacent: they share vertex $B$ and side $\overrightarrow{BD}$ . But $\angle ABD$ and $\angle ABC$ are not adjacent, because the smaller one sits inside the larger one.

Angles ABD and DBC are adjacent. Together they form a right angle, so they are also complementary: 35° + 55° = 90°.

Complementary and supplementary angles

Definition

Complementary and supplementary

Two angles are complementary if their measures add up to $90^\circ$ . Two angles are supplementary if their measures add up to $180^\circ$ .

Each angle is called the complement (or supplement) of the other. The complement of $62^\circ$ is $90 - 62 = 28^\circ$ , and its supplement is $180 - 62 = 118^\circ$ . An angle of $x^\circ$ has complement $(90 - x)^\circ$ and supplement $(180 - x)^\circ$ .

Common mistake

Complementary and supplementary angles do not have to be adjacent or even near each other. A $30^\circ$ angle in one corner of the page and a $60^\circ$ angle in another are complementary. The definitions care only about the sum. Also, only acute angles have complements, since an angle of $90^\circ$ or more leaves nothing to add.

Tip

To keep them straight: c comes before s in the alphabet, and $90$ comes before $180$ . Or think "Corner" for $90^\circ$ and "Straight" for $180^\circ$ .

Linear pairs and vertical angles

When two lines intersect, they form four angles.

Two intersecting lines form four angles. Angles 1 and 3 are vertical angles, as are angles 2 and 4. Each pair of neighbors, such as 1 and 2, is a linear pair.

A linear pair is two adjacent angles whose outside sides are opposite rays. Together they form a straight line, so they fill a straight angle. In the figure, $\angle 1$ and $\angle 2$ form a linear pair, and so do $\angle 2$ and $\angle 3$ , $\angle 3$ and $\angle 4$ , and $\angle 4$ and $\angle 1$ .

Vertical angles are two angles whose sides form two pairs of opposite rays. They are the angles directly across from each other at the crossing: $\angle 1$ and $\angle 3$ , and $\angle 2$ and $\angle 4$ .

Two facts about intersecting lines

Linear Pair Postulate: the angles of a linear pair are supplementary.
Vertical Angles Theorem: vertical angles are congruent.

You don't have to take the second fact on faith. It follows from the first. Angles $1$ and $2$ form a linear pair, and so do angles $2$ and $3$ :

$m\angle 1 + m\angle 2 = 180 \qquad \text{and} \qquad m\angle 2 + m\angle 3 = 180.$

Both sums equal $180$ , so $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$ . Subtract $m\angle 2$ from both sides, and $m\angle 1 = m\angle 3$ . The same argument works for $\angle 2$ and $\angle 4$ . This is your first real geometric proof, and you'll write it formally in the next unit.

Examples

Worked example: All four angles from one

In the intersecting lines above, $m\angle 1 = 40^\circ$ . Find the other three angles.

$\angle 3$ is vertical to $\angle 1$ , so $m\angle 3 = 40^\circ$ .
$\angle 2$ forms a linear pair with $\angle 1$ , so $m\angle 2 = 180 - 40 = 140^\circ$ .
$\angle 4$ is vertical to $\angle 2$ , so $m\angle 4 = 140^\circ$ .

Check: $40 + 140 + 40 + 140 = 360$ , a full turn around the point.

Worked example: Vertical angles with algebra

Two vertical angles measure $(5x - 12)^\circ$ and $(3x + 20)^\circ$ . Find $x$ , the measure of each angle, and the measure of an angle adjacent to them.

Vertical angles are congruent, so $5x - 12 = 3x + 20$ . Then $2x = 32$ and $x = 16$ . Each angle measures $5(16) - 12 = 68^\circ$ . An adjacent angle forms a linear pair with one of them, so it measures $180 - 68 = 112^\circ$ .

Worked example: A linear pair with algebra

Two angles form a linear pair. They measure $(4x + 8)^\circ$ and $(2x - 2)^\circ$ . Find both angles.

A linear pair is supplementary:

\begin{aligned} (4x + 8) + (2x - 2) &= 180 \\ 6x + 6 &= 180 \\ x &= 29 \end{aligned}

The angles are $4(29) + 8 = 124^\circ$ and $2(29) - 2 = 56^\circ$ . Check: $124 + 56 = 180$ .

Worked example: A word problem

An angle measures $30^\circ$ less than twice its supplement. Find the angle.

Let the angle be $x$ . Its supplement is $180 - x$ . "Thirty less than twice the supplement" is $2(180 - x) - 30$ . So

\begin{aligned} x &= 2(180 - x) - 30 \\ x &= 330 - 2x \\ 3x &= 330 \\ x &= 110 \end{aligned}

The angle is $110^\circ$ . Check: its supplement is $70^\circ$ , and $2(70) - 30 = 110$ .