We talk a lot about degree measure in geometry: a right angle is 90 degrees, the angles in a triangle make 180 degrees, and on and on. But just what is a degree? Let’s give this term a precise definition, and then let’s use the symmetry of the circle to show that the measure of a straight angle must be 180 degrees.

The degree is a unit of measure that can be used to describe the magnitude of a rotation. We’d like to be able to say that a turn through 360 degrees is a full turn; this will be the basis for our definition.

Beginning in Grade 4, students learn to work with the degree as a unit of measure for angles. In Standard 4.MD.5, we read this:

An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

[Note that the distance around a circular arc is an undefined notion in the CCSS, per Standard G-CO.1.]

As an example, the angle below cuts an arc that is 40/360 as long as the complete circle, and so the degree measure of $\angle AOB$ is 40.

We apply this understanding each time we use a protractor to measure an angle, as the sides of the angle cut part of the circle at the boundary of the protractor.

This brings us to an interesting question: if two students use protractors of different sizes to measure the same angle, will they always get the same measurement? For instance, if the purple arc in the figure below is 40/360 of the circle it lives on, does that necessarily mean that the red arc is 40/360 of the circle it lives on? This question will be taken up in the lessons on size transformations.

Now that we have a clear idea of what it means to measure an angle, let’s state an important theorem.

Straight Angle Theorem. Let $A$ and $B$ be points on a circle with center $O$. Then the measure of arc $AB$ is $180^\circ$ if and only if points $A$, $O$, and $B$ are collinear.

This theorem can be proven by showing that the reflection over line $AB$ maps the arc above the line to the arc below. Since a reflection preserves angles, the two arcs must be equal, and so each arc must be $180^\circ$.

Next up: let’s use the Straight Angle Theorem to establish the Vertical Angles Theorem using a rotation.