Math Core

When lines cross or angles sit side by side, their measures are connected. If you know one angle, you can often find several others without a protractor, just by writing and solving a simple equation.

Angle pairs with special sums

Two angles are adjacent if they share a vertex and a side but don't overlap. Adjacent angles often combine to make a right angle or a straight line, and those give you two important kinds of pairs.

Definition

Complementary and supplementary angles

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

A handy memory trick: "C" comes before "S" in the alphabet, and $90$ comes before $180$ .

A right angle split into two adjacent angles. The two parts are complementary.

Worked example: Finding a complement

In the figure, a right angle is split into a $35^\circ$ angle and an angle of $x^\circ$ . Find $x$ .

The two angles are complementary, so

\begin{aligned} x + 35 &= 90 \\ x &= 55 \end{aligned}

Vertical angles

When two lines cross, they make four angles. The angles directly across from each other are called vertical angles.

Two intersecting lines. Angles a and c are vertical angles, and so are b and d.

Angles $a$ and $b$ sit side by side on a straight line, so $a + b = 180^\circ$ . Angles $b$ and $c$ also sit on a straight line, so $b + c = 180^\circ$ . Both $a$ and $c$ are " $180^\circ$ minus $b$ ," so they must be equal.

Vertical angles are equal

When two lines intersect, vertical angles have equal measures, and any two neighboring angles are supplementary.

In the figure, $a = c$ and $b = d$ . If $a = 60^\circ$ , then $c = 60^\circ$ and $b = d = 180^\circ - 60^\circ = 120^\circ$ .

Writing equations for unknown angles

Most angle problems follow three steps:

Decide how the angles are related (equal, add to $90^\circ$ , or add to $180^\circ$ ).
Write an equation.
Solve, then answer the question that was actually asked.

Two adjacent angles along a straight line are supplementary.

Worked example: Supplementary angles with expressions

In the figure, the two angles on the straight line measure $(2x + 10)^\circ$ and $(3x - 5)^\circ$ . Find both angles.

The angles make a straight line, so they add to $180^\circ$ :

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

The angles are $2(35) + 10 = 80^\circ$ and $3(35) - 5 = 100^\circ$ . Check: $80 + 100 = 180$ .

Worked example: Vertical angles with expressions

Two vertical angles measure $(4x - 12)^\circ$ and $(2x + 30)^\circ$ . Find the measure of each angle.

Vertical angles are equal:

\begin{aligned} 4x - 12 &= 2x + 30 \\ 2x &= 42 \\ x &= 21 \end{aligned}

Each angle measures $4(21) - 12 = 72^\circ$ . Check: $2(21) + 30 = 72$ .

Common mistake

Finding $x$ is often not the last step. If a question asks for the angle, substitute $x$ back into the expression. In the last example, $x = 21$ , but the angle is $72^\circ$ .

Tip

Before solving, estimate from the picture. An angle that looks a little less than a right angle should come out a bit under $90^\circ$ . If you get $150^\circ$ , recheck your equation.