Angles and their parts

Open a pair of scissors. Open a book. Open your arms wide. In each case you have made an angle. Angles tell us how much one direction turns from another, and they are everywhere in geometry, art, sport, and engineering.

Concept

An angle is formed when two rays share the same starting point. The shared starting point is called the vertex, and the two rays are called the arms (or sides) of the angle. We name an angle in one of three ways:

• by its vertex alone, like $\angle B$, when there is no chance of confusion;
• by three points $\angle ABC$, where $B$ is the vertex and $A$ and $C$ are points on the arms (the vertex always goes in the middle);
• by a small letter or number placed inside the angle, like $\angle 1$ or $\angle x$.

Notice that $\angle ABC$ and $\angle CBA$ refer to the same angle , order of the outer two letters does not matter, as long as the vertex letter is in the middle.

Every angle divides the plane into three parts: the angle's interior (inside the opening), the exterior (outside), and the angle itself (the two rays). Knowing which region is which is important when we shade or measure.

The size of an angle is not about how long its arms are. If you draw $\angle ABC$ with arms $5$ cm long, and a friend draws the same angle with arms $15$ cm long, they are the same angle, because they describe the same amount of turn. What matters is the spread between the arms.

To compare two angles without measuring, you can place one on top of the other so their vertices coincide and one arm matches up. Whichever angle's other arm is "more open" is the larger angle. This is the original way angles were compared, long before protractors existed.

A zero angle is what you get when both rays point in exactly the same direction, like a closed pair of scissors. A straight angle is what you get when the two rays point in exactly opposite directions, forming a straight line. Every other angle lies somewhere in between.

Worked examples

Example 1. In $\angle PQR$, name the vertex and the two arms.

• Vertex: $Q$ (the middle letter).
• Arms: ray $\overrightarrow{QP}$ and ray $\overrightarrow{QR}$.

Example 2. Suppose four rays $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$ all start at $O$. How many distinct angles are formed?

• We pick any two rays. Number of pairs $= \binom{4}{2} = 6$.
• So $6$ angles: $\angle AOB, \angle AOC, \angle AOD, \angle BOC, \angle BOD, \angle COD$.

Example 3. A clock shows $3$ o'clock. The minute hand points to $12$ and the hour hand points to $3$. The angle between them has its vertex where?

• At the centre of the clock, where both hands meet.

Example 4. Two angles are drawn. $\angle X$ has arms of length $4$ cm. $\angle Y$ is the same opening but with arms of length $9$ cm. Which is larger?

• Neither , they are the same angle. Arm length does not change the measure of an angle.

Try it yourself

1. Draw $\angle PQR$ where $Q$ is the vertex. Label the arms.
2. Rename $\angle DEF$ in another correct form.
3. Five rays come out of point $O$. How many angles are formed?
4. Sketch a zero angle and a straight angle side by side.
5. A pair of scissors is open to form an angle. What is the vertex?
6. Look at the corner of your textbook. Identify the vertex of the angle and the two arms.
7. Is the angle made by your arms when you stretch them out wide bigger than when they point straight up? Why?

Activity

Paper-fan angles. Take a strip of paper and fold it like a hand-fan. Pick out one fold and find the vertex of the angle, the two arms, and the interior. Now open the fan wide and slowly close it. As you close, the angle gets smaller and smaller until it reaches zero. Identify two positions where the angle is roughly the same as a corner of your notebook (a right angle).