Angles and Their Measure

Name the rays and the opening

An angle is formed by two rays with a common endpoint, called its vertex. If the rays are $\overrightarrow{OA}$ and $\overrightarrow{OB}$, their vertex is $O$. Name the angle $\angle AOB$ or $\angle BOA$: the vertex goes in the middle. The outer letters name points on the two rays, not endpoints where the rays stop.

The rays bound two openings: one on either side. Here the short arc selects the smaller opening. Unless a different opening is marked or named, three-letter angle notation means the opening of at most a half turn. To ask for the larger opening, we will say reflex explicitly. You may shorten the name to $\angle O$ only when the intended angle at $O$ is unambiguous.

Degrees count the turn

A degree is $\frac{1}{360}$ of a full turn. Write $65^\circ$ for 65 degrees. The notation $m\angle AOB = 65^\circ$ says that the measure of angle $AOB$ is 65 degrees. The angle is the geometric object; its measure is the number with the degree unit.

A full turn measures $360^\circ$ by this choice of unit. We accept angle addition, the additivity of angle measures, as a measurement foundation: the measures of consecutive parts with nonoverlapping interiors that fill a selected opening or turn exactly once add to the whole measure. This includes parts that fill a full turn. The selected route around the vertex and its parts must be specified.

A quarter turn is $360^\circ \div 4 = 90^\circ$. A half turn is $180^\circ$. For example, $35^\circ$ is less than a quarter turn, while $115^\circ$ lies between a quarter turn and a half turn. The names below describe the selected opening.

A $90^\circ$ right angle is the square corner used to describe perpendicular lines in lesson 1.1.

A $0^\circ$ turn makes no rotation. Its final ray points in the same direction as its starting ray, just as after a full turn, but the turns are different. Their directions alone do not record how far you turned.

Take two rays with different directions that bound a selected $125^\circ$ opening. The other route between the rays completes one full turn with that opening. By the accepted additivity foundation, the other opening measures

$360^\circ - 125^\circ = 235^\circ.$

This is between a half turn and a full turn, so it is the reflex opening.

For any two rays with different directions, following their two openings around the vertex fills one full turn with no overlapping interiors. Additivity and the degree unit make the two measures total $360^\circ$, so subtracting either measure from $360^\circ$ gives the other. When the smaller opening is strictly between $0^\circ$ and $180^\circ$, the other is reflex. For opposite rays, both openings are $180^\circ$ instead.

Change the turn below, then change each drawn ray length while keeping the turn fixed. Try $90^\circ$, $180^\circ$, $235^\circ$, and $360^\circ$. The full-circle protractor counts counterclockwise from the fixed blue ray; it has a different layout from the semicircular protractor used in the next section.

Use a protractor from its zero

A protractor is a tool with a degree scale for measuring or drawing angles. Its center mark belongs on the vertex. Its straight baseline must lie along one ray. Read the scale whose $0$ lies on that starting ray, following that scale to the other ray. If a drawn ray is too short to reach the scale, extend its drawing along the same direction. That does not change the angle.

To measure a reflex opening with a semicircular protractor, measure the smaller opening first and subtract it from $360^\circ$. This works because the two selected openings make one full turn.

Add parts only when they fill the opening

Suppose a ray inside a $100^\circ$ opening cuts it into parts of $35^\circ$ and $65^\circ$. Following the first turn and then the second reaches the same final ray as the whole turn:

$35^\circ + 65^\circ = 100^\circ.$

Apply the angle-addition foundation already stated in the degree section. When $\overrightarrow{OB}$ lies inside the selected angle $AOC$ and its two selected parts fill that opening without overlapping interiors,

$m\angle AOB + m\angle BOC = m\angle AOC.$

The parts must fill the selected opening exactly once. Merely sharing a vertex is not enough. For a reflex opening, specify that opening explicitly and follow its parts around it. For three consecutive parts, combine the first two into one angle, then apply angle addition again to combine that angle with the third part.

A bisector makes equal angular parts

An angle bisector is a ray from the vertex inside the selected opening that divides it into two angles of equal measure. For a $130^\circ$ opening, let each part measure $x$ degrees. The parts are equal by the bisector definition and add to the whole by angle addition:

$x + x = 130.$

Thus $2x = 130$, so each part measures

$130^\circ \div 2 = 65^\circ.$

Angles with equal measures are called congruent angles. Write $\angle AOB \cong \angle BOC$ for the angles, and $m\angle AOB = m\angle BOC$ for their measures. Congruence markings on a diagram record that equality; they do not follow merely from the way the sketch looks. In the diagram below, the single tick on each angle arc is the matching mark. Equal drawn ray lengths are not required.

In the explorer, set the turn to $130^\circ$ and show the bisector. Then change the drawn ray lengths. Each part remains $65^\circ$ because the bisector divides the turn. For $0^\circ$ and $360^\circ$, this explorer leaves the bisector hidden.

The reference from lessons 1.1 and 1.2 now gains the following entries.

These results follow from the turn and bisector definitions, the measurement foundations, and arithmetic. They do not require later angle-pair or triangle theorems.

Common mistakes