Math Core

Segments measure how far apart two points are. Angles measure something different: how much you turn. You'll meet angles in every unit that follows, so it's worth getting the names, the notation and the two measuring postulates exactly right now.

What an angle is

Definition

Angle

An angle is made of two rays that share an endpoint. The rays are the sides of the angle, and the shared endpoint is its vertex.

Angle ABC has vertex B and sides BA and BC. Its measure is 50°.

You can name the angle in the figure three ways:

with three points, vertex in the middle: $\angle ABC$ or $\angle CBA$ ;
with the vertex alone: $\angle B$ , but only when no other angle shares that vertex;
with a number written inside the angle: $\angle 1$ .

The measure of an angle, written $m\angle ABC$ , is how far one side is turned from the other, in degrees. A full turn is $360^\circ$ , so a half turn is $180^\circ$ . Just as with segments, $\angle ABC$ is the figure and $m\angle ABC$ is a number: you write $m\angle ABC = 50^\circ$ .

Measuring with a protractor

A protractor works like a curved ruler. Place its center on the vertex, and each side of the angle passes through a number from $0$ to $180$ . This idea is the Protractor Postulate: the measure of the angle is the absolute value of the difference of the two readings. If one side passes through $30$ and the other through $110$ , the angle measures $|110 - 30| = 80^\circ$ . You don't have to line a side up with $0$ .

Classifying angles

Type	Measure
acute	between $0^\circ$ and $90^\circ$
right	exactly $90^\circ$
obtuse	between $90^\circ$ and $180^\circ$
straight	exactly $180^\circ$ (the sides are opposite rays)

An acute, a right, an obtuse and a straight angle. The small square marks a right angle.

In a diagram, a small square in the corner means the angle is exactly $90^\circ$ . Don't assume an angle is right just because it looks right.

Adding angles

Angles combine the same way segments do. If a ray splits an angle into two smaller angles, the two parts add up to the whole.

Ray BD is in the interior of angle ABC, so m∠ABD + m∠DBC = m∠ABC.

Angle Addition Postulate

If point $D$ is in the interior of $\angle ABC$ , then

$m\angle ABD + m\angle DBC = m\angle ABC.$

Two angles are congruent if they have the same measure: $\angle 1 \cong \angle 2$ means $m\angle 1 = m\angle 2$ . Matching arcs in a diagram mark congruent angles. A ray that divides an angle into two congruent angles is the angle bisector. If $\overrightarrow{BD}$ bisects $\angle ABC$ , then each half is $\tfrac12 m\angle ABC$ .

Common mistake

When three rays share a vertex, the name $\angle B$ is ambiguous: in the figure above, it could mean $\angle ABD$ , $\angle DBC$ or $\angle ABC$ . Use three letters whenever more than one angle has the same vertex, and always put the vertex letter in the middle.

Examples

Worked example: Using a protractor

The center of a protractor is on vertex $B$ . Side $\overrightarrow{BA}$ passes through $130$ and side $\overrightarrow{BC}$ passes through $45$ . Find $m\angle ABC$ and classify the angle.

$m\angle ABC = |130 - 45| = 85^\circ$ . Since $85^\circ$ is between $0^\circ$ and $90^\circ$ , the angle is acute.

Worked example: Angle addition with algebra

$D$ is in the interior of $\angle ABC$ , with $m\angle ABD = (3x + 4)^\circ$ , $m\angle DBC = (2x + 11)^\circ$ and $m\angle ABC = 90^\circ$ . Find both parts.

By the Angle Addition Postulate,

\begin{aligned} (3x + 4) + (2x + 11) &= 90 \\ 5x + 15 &= 90 \\ x &= 15 \end{aligned}

So $m\angle ABD = 3(15) + 4 = 49^\circ$ and $m\angle DBC = 2(15) + 11 = 41^\circ$ . Check: $49 + 41 = 90$ .

Worked example: An angle bisector

$\overrightarrow{BD}$ bisects $\angle ABC$ , with $m\angle ABD = (5x - 8)^\circ$ and $m\angle DBC = (3x + 14)^\circ$ . Find $m\angle ABC$ and classify it.

A bisector makes two congruent angles, so $5x - 8 = 3x + 14$ . Then $2x = 22$ and $x = 11$ . Each half is $5(11) - 8 = 47^\circ$ , so $m\angle ABC = 2 \cdot 47 = 94^\circ$ . That is between $90^\circ$ and $180^\circ$ , so $\angle ABC$ is obtuse, even though each half is acute.

Tip

After solving for $x$ , check every angle measure you found. Each part must be positive, and the parts must add up to the whole. If a part comes out negative or larger than the whole, go back and find the error.

Practice

Practice 1

An angle measures $179^\circ$ . How is it classified?

Practice 2

Three rays $\overrightarrow{KJ}$ , $\overrightarrow{KL}$ and $\overrightarrow{KM}$ share the endpoint $K$ . Which name can not refer to an angle formed by these rays?

Practice 3

A protractor's center is on the vertex of an angle. One side passes through $20$ and the other passes through $145$ . Find the measure of the angle in degrees.