Types of Angles

An angle is formed when two rays share a common starting point, called the vertex. Angles are measured in degrees (written with the symbol $\degree$), and a full rotation around a point is $360\degree$. Understanding angle types is the first step in geometry — every cut, bend, and joint you encounter on a job site starts with knowing your angles.

Measuring Angles in Degrees

Think of degrees as slices of a circle. A full circle is $360\degree$. A half circle (a straight line) is $180\degree$. A quarter circle (a square corner) is $90\degree$.

Angles are measured using a protractor, and they are classified by their size.

Types of Angles

Acute Angle

An acute angle measures less than $90\degree$.

$0\degree < \text{acute angle} < 90\degree$

Examples: $30\degree$, $45\degree$, $60\degree$, $89\degree$

Think of a slightly open door — the angle between the door and the wall is acute.

Right Angle

A right angle measures exactly $90\degree$.

$\text{right angle} = 90\degree$

Right angles are marked with a small square symbol at the vertex. They are everywhere in construction — walls meeting floors, corners of rooms, and square cuts on lumber.

Obtuse Angle

An obtuse angle measures more than $90\degree$ but less than $180\degree$.

$90\degree < \text{obtuse angle} < 180\degree$

Examples: $100\degree$, $120\degree$, $150\degree$, $179\degree$

Think of a door opened past the halfway point — the angle between the door and the wall on the hinge side is obtuse.

Straight Angle

A straight angle measures exactly $180\degree$.

$\text{straight angle} = 180\degree$

It looks like a straight line. The two rays point in exactly opposite directions.

Reflex Angle

A reflex angle measures more than $180\degree$ but less than $360\degree$.

$180\degree < \text{reflex angle} < 360\degree$

Examples: $200\degree$, $270\degree$, $350\degree$

Reflex angles are measured by going “the long way around” from one ray to the other.

Angle Classification Summary

Complementary Angles

Two angles are complementary when their measures add up to $90\degree$.

$\text{angle } A + \text{angle } B = 90\degree$

Example 1: Find the complement of $35\degree$

$90\degree - 35\degree = 55\degree$

Answer: The complement of $35\degree$ is $55\degree$. Together, $35\degree + 55\degree = 90\degree$.

Example 2: Two angles are complementary. One measures $22\degree$. Find the other.

$90\degree - 22\degree = 68\degree$

Answer: The other angle is $68\degree$.

Key fact: Both complementary angles must be acute (less than $90\degree$), because if either one were $90\degree$ or more, the pair could never sum to exactly $90\degree$.

Supplementary Angles

Two angles are supplementary when their measures add up to $180\degree$.

$\text{angle } A + \text{angle } B = 180\degree$

Example 3: Find the supplement of $110\degree$

$180\degree - 110\degree = 70\degree$

Answer: The supplement of $110\degree$ is $70\degree$. Together, $110\degree + 70\degree = 180\degree$.

Supplementary angles often appear along a straight line. If a straight line is split by a ray, the two angles on either side are supplementary.

Real-World Application: Carpentry — Miter Cuts

When a carpenter builds a picture frame or installs crown molding, they join two pieces of wood at a corner. A standard corner is $90\degree$, and the carpenter needs to cut each piece at the correct angle so the two pieces meet flush.

Problem: You need two pieces of baseboard to meet at a $90\degree$ corner. What angle should each miter cut be?

Step 1: The total corner angle is $90\degree$.

Step 2: A miter joint splits the corner angle equally between the two pieces:

$\frac{90\degree}{2} = 45\degree$

Step 3: Set your miter saw to $45\degree$ and cut each board.

Answer: Each board gets a $45\degree$ cut. The two $45\degree$ cuts meet to form the $90\degree$ corner.

This same logic applies to other angles. For a $120\degree$ corner (common in hexagonal frames), each miter cut would be:

$\frac{120\degree}{2} = 60\degree$

Getting the angle wrong by even a degree or two creates visible gaps in the joint — this is why understanding angles matters in finish carpentry.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Angles are measured in degrees, with a full rotation being $360\degree$
• The five angle types are acute ($<90\degree$), right ($=90\degree$), obtuse ($>90\degree$), straight ($=180\degree$), and reflex ($>180\degree$)
• Complementary angles add to $90\degree$; supplementary angles add to $180\degree$
• Miter cuts split a corner angle in half — a $90\degree$ corner needs two $45\degree$ cuts
• Right angles are the most common angle in construction, but knowing all angle types is essential for accurate work

Return to Geometry for more topics in this section.