Types of angles

Just as we give names to people, we give names to angles based on how big they are. Knowing the name of an angle the moment you see it is a useful geometric reflex.

Concept

There are six standard names for angles based on their measure in degrees:

A right angle is the most important angle in geometry. We mark it with a small square at the vertex (not an arc). Right angles appear in squares, rectangles, the edges of bricks, and most everyday surfaces because builders and carpenters work with right angles all the time.

An acute angle is anything smaller than a right angle , for example $30°$ or $75°$ , and feels "sharp" or "pointy". An obtuse angle is bigger than a right angle but smaller than a straight angle , for example $120°$ or $150°$. These angles look "wide and open".

A straight angle is exactly $180°$. The two arms point in opposite directions and lie on a single line, so it looks like there is no angle at all , but in geometry it counts.

A reflex angle is the surprising one. If you measure the angle on the outside of an acute or obtuse angle, you get a reflex angle. For instance, if an angle is $70°$ on one side, then on the other side it is $360° - 70° = 290°$, a reflex angle. Both numbers describe the same drawing , just measured "the long way round".

A complete angle ($360°$) is one full turn. After spinning $360°$ you are facing the same direction you started.

Worked examples

Example 1. Classify the angle of measure $115°$.

• $90° < 115° < 180°$, so it is obtuse.

Example 2. $\angle ABC = 30°$. What is the reflex angle of $\angle ABC$?

• Reflex $= 360° - 30° = 330°$.

Example 3. The angle at the corner of a chessboard. What type?

• A corner of a chessboard is $90°$ , a right angle.

Example 4. At $7{:}00$, what type of angle do the hands of a clock make?

• From $12$ to $7$ is $5$ parts of $30°$ each $= 150°$ , an obtuse angle. (The smaller of the two angles between the hands.)
• The reflex side is $360° - 150° = 210°$.

Try it yourself

1. Classify these angles: $45°$, $90°$, $130°$, $180°$, $270°$, $360°$.
2. The reflex angle of $80°$ is what?
3. Sketch one angle of each type listed (acute, right, obtuse, straight, reflex).
4. The hands of a clock at $9{:}00$ make what angle? What type?
5. Are the angles in an equilateral triangle (where every angle is $60°$) acute, right, or obtuse?
6. Walk around in a square. How many right angles do you turn in total? What is the sum of those angles?
7. Tricky: If an angle equals its own complement plus its own supplement, find it. (Hint: complement of $\theta$ is $90° - \theta$, supplement is $180° - \theta$. Equation: $\theta = (90° - \theta) + (180° - \theta)$.)

Activity

Angle scavenger hunt. With your protractor and a notebook, walk around your home for five minutes. Find and sketch at least:

• $3$ acute angles
• $3$ right angles
• $2$ obtuse angles
• $1$ straight angle

For each, write where you found it (e.g. "open scissors", "open window", "corner of TV"). Compare with a classmate next class.