Table of Contents
Measuring and Describing Angles
In the parent chapter you met the basic idea of an angle as the “opening” between two rays with a common endpoint. Here we focus on the different types of angles and how we classify them.
Throughout this chapter, we will use degrees to measure angles. A full turn around a point is $360^\circ$.
Basic Types of Angles by Size
We classify angles according to how many degrees they measure.
Zero Angle
A zero angle has measure $0^\circ$.
Both rays lie exactly on top of each other and there is no visible “opening.” You can think of starting to rotate a ray but not moving it at all.
Acute Angle
An acute angle is smaller than a right angle.
Definition:
- An angle is acute if its measure is between $0^\circ$ and $90^\circ$.
In symbols:
- Acute angle: $0^\circ < \theta < 90^\circ$.
Examples:
- $30^\circ$, $45^\circ$, $60^\circ$ are all acute angles.
Right Angle
A right angle is exactly one quarter of a full turn.
Definition:
- An angle is a right angle if its measure is exactly $90^\circ$.
In symbols:
- Right angle: $\theta = 90^\circ$.
Right angles are often marked with a small square at the vertex instead of a curved arc.
Obtuse Angle
An obtuse angle is bigger than a right angle but smaller than a straight angle.
Definition:
- An angle is obtuse if its measure is between $90^\circ$ and $180^\circ$.
In symbols:
- Obtuse angle: $90^\circ < \theta < 180^\circ$.
Example:
- A $120^\circ$ angle is obtuse.
Straight Angle
A straight angle forms a straight line.
Definition:
- An angle is a straight angle if its measure is exactly $180^\circ$.
In symbols:
- Straight angle: $\theta = 180^\circ$.
The two rays point in exactly opposite directions and lie on the same line.
Reflex Angle
Reflex angles are larger than a straight angle but less than a full turn.
Definition:
- An angle is reflex if its measure is between $180^\circ$ and $360^\circ$.
In symbols:
- Reflex angle: $180^\circ < \theta < 360^\circ$.
Example:
- A $270^\circ$ angle is reflex.
Full Angle (Complete Angle)
A full angle is a complete turn around a point.
Definition:
- A full angle (or complete angle) has measure $360^\circ$.
In symbols:
- Full angle: $\theta = 360^\circ$.
After turning $360^\circ$, the ray points in the same direction as where it started.
Relationships Between Angles
Besides size, we also classify angles by how they relate to each other. In this section we focus on relationships between pairs of angles that share a specific numerical connection.
Complementary Angles
Complementary angles “add up to a right angle.”
Definition:
- Two angles are complementary if the sum of their measures is $90^\circ$.
If the two angles are $\alpha$ and $\beta$, then they are complementary when
$$
\alpha + \beta = 90^\circ.
$$
Each angle is called the complement of the other.
Examples:
- $30^\circ$ and $60^\circ$ are complementary.
- $50^\circ$ and $40^\circ$ are complementary.
Note: Complementary angles do not have to be next to each other. They only need their measures to add to $90^\circ$.
Supplementary Angles
Supplementary angles “add up to a straight angle.”
Definition:
- Two angles are supplementary if the sum of their measures is $180^\circ$.
If the two angles are $\alpha$ and $\beta$, then they are supplementary when
$$
\alpha + \beta = 180^\circ.
$$
Each angle is called the supplement of the other.
Examples:
- $110^\circ$ and $70^\circ$ are supplementary.
- $45^\circ$ and $135^\circ$ are supplementary.
Again, supplementary angles do not have to share a side; only their measures must add to $180^\circ$.
Adjacent Angles
Adjacent angles share a side and a vertex.
Definition:
- Two angles are adjacent if:
- they share a common vertex,
- they share exactly one common side (ray),
- their interiors do not overlap.
Adjacent angles “sit next to” each other and touch along one ray.
Example:
- If angle $AOB$ is $30^\circ$ and angle $BOC$ is $60^\circ$, and they share the side $OB$, then $\angle AOB$ and $\angle BOC$ are adjacent.
Adjacent angles are often used to form complementary or supplementary pairs:
- Two adjacent angles that are complementary form a right angle together.
- Two adjacent angles that are supplementary form a straight angle together.
Linear Pair of Angles
A linear pair is a special kind of adjacent angles.
Definition:
- Two angles form a linear pair if:
- they are adjacent, and
- their non-common sides form a straight line.
Since the non-common sides form a straight line, the two angles in a linear pair are always supplementary:
$$
\alpha + \beta = 180^\circ.
$$
Vertical (Opposite) Angles
Vertical angles are formed when two lines intersect.
Definition:
- When two lines intersect, they create two pairs of vertical angles (also called opposite angles). Each pair consists of two angles that are opposite each other and share only a vertex (no common sides).
Key fact (used heavily later in geometry):
- Vertical angles always have equal measures.
If two lines intersect and form angles $\alpha$ and $\beta$ as a vertical pair, then
$$
\alpha = \beta.
$$
Example:
- If one angle at an intersection measures $40^\circ$, the vertical angle across from it also measures $40^\circ$.
Classifying Angles in Practice
When you see an angle or a pair of angles, you can describe them in several ways at once.
For a single angle:
- Name it by its measure type:
- zero, acute, right, obtuse, straight, reflex, or full.
For a pair of angles:
- Look at both their size and their relationship:
- Do they share a vertex and a side? They may be adjacent.
- Do they form a straight line together? They may be a linear pair and therefore supplementary.
- Are they across from each other at an intersection of two lines? They are vertical angles and equal in measure.
- Do their measures add to $90^\circ$ or $180^\circ$? Then they are complementary or supplementary, respectively.
Recognizing these angle types is essential for solving geometric problems, especially when you need to find unknown angle measures in figures with intersecting or parallel lines and polygons.