Math Core

Angles follow a few simple rules: some pairs always add to $90^\circ$ , some always add to $180^\circ$ , and the three angles of any triangle always add to $180^\circ$ . Each rule hands you an equation, so you can use the equation-solving skills you already have to find a missing angle.

Angle pairs

An angle is measured in degrees. A right angle is $90^\circ$ and a straight line makes a $180^\circ$ angle. Angles smaller than $90^\circ$ are acute, and angles between $90^\circ$ and $180^\circ$ are obtuse.

Three kinds of angle pairs come up all the time:

Complementary angles add to $90^\circ$ . Together they make a right angle.
Supplementary angles add to $180^\circ$ . Two angles side by side on a straight line are supplementary.
Vertical angles are the opposite angles formed where two lines cross. Vertical angles are always equal.

Worked example: An equation from vertical angles

Two lines cross. One angle measures $(3x + 12)^\circ$ and the angle directly across from it measures $(5x - 20)^\circ$ . Find $x$ and the size of each angle.

Vertical angles are equal, so set the expressions equal and solve:

\begin{aligned} 3x + 12 &= 5x - 20 \\ 12 &= 2x - 20 \\ 32 &= 2x \\ x &= 16 \end{aligned}

Each angle is $3(16) + 12 = 60^\circ$ . Check with the other expression: $5(16) - 20 = 60^\circ$ . ✓

The angles of a triangle

Draw any triangle, tear off its three corners, and line them up point to point. They always fit together to form a straight line. That's why the three angles of every triangle add to $180^\circ$ , whether the triangle is tall and skinny or short and wide.

Triangle angle sum

In every triangle, the three interior angles add to $180^\circ$ :

\angle A + \angle B + \angle C = 180^\circ

A triangle with two known angles.

Worked example: Finding the third angle

Find $x$ in the triangle above.

\begin{aligned} 60 + 40 + x &= 180 \\ 100 + x &= 180 \\ x &= 80 \end{aligned}

The third angle is $80^\circ$ .

Classifying triangles

You can name a triangle by its angles:

Acute triangle: all three angles are acute.
Right triangle: one angle is exactly $90^\circ$ .
Obtuse triangle: one angle is obtuse.

Or by its sides:

Equilateral: all three sides are equal (and all three angles are $60^\circ$ ).
Isosceles: at least two sides are equal. The angles opposite the equal sides, called the base angles, are equal too.
Scalene: no sides are equal.

A triangle can have at most one right angle or one obtuse angle. Two of them would already add to $180^\circ$ or more, leaving nothing for the third angle.

Worked example: Angles written with a variable

The angles of a triangle measure $x^\circ$ , $2x^\circ$ and $3x^\circ$ . Find each angle and classify the triangle by its angles.

\begin{aligned} x + 2x + 3x &= 180 \\ 6x &= 180 \\ x &= 30 \end{aligned}

The angles are $30^\circ$ , $60^\circ$ and $90^\circ$ . One angle is $90^\circ$ , so it's a right triangle.

Exterior angles

Extend one side of a triangle past a corner. The angle between the extension and the neighboring side is an exterior angle. It sits on a straight line next to an interior angle, so the two are supplementary.

The exterior angle y° is next to the interior angle at the bottom right corner.

Worked example: An exterior angle

Find $y$ in the figure above.

The interior angle at the bottom right is $180 - 60 - 80 = 40^\circ$ . The exterior angle is supplementary to it: $y = 180 - 40 = 140$ .

Notice that $140 = 60 + 80$ . That's no accident.

Tip

An exterior angle always equals the sum of the two interior angles not next to it. In the figure, $y = 60 + 80 = 140$ , with no need to find the third angle first.

Common mistake

Don't mix up the two sums. Complementary angles add to $90^\circ$ ; supplementary angles add to $180^\circ$ . A memory trick: c comes before s in the alphabet, and $90$ comes before $180$ .