Adjacent Angles: Varieties and Examples

Adjacent angles are a fundamental concept in basic geometry, serving as a foundation for more advanced topics like trigonometry, physics, and engineering. Geometry students should be proficient in identifying adjacent angles and calculating them with high accuracy, even without the aid of a protractor.

How to Spot Adjacent Angles

Simply put, adjacent angles are two angles that share a common vertex or side and do not overlap each other.

Understanding the value of one or more adjacent angles can assist in determining unknown angles, thanks to the properties of adjacent angles and their relationships. For instance, knowing the sum of all angles with a common vertex can simplify the process of calculating adjacent angles.

Although the simplest adjacent angles consist of just two angle values, multiple pairs of adjacent angles can be adjacent to each other, forming relationships that add up to sums reaching, but never exceeding, 360 degrees.

3 Types of Adjacent Angles

1. Supplementary Angles

Supplementary angles are adjacent angles that, when combined, total 180 degrees (or π radians). These angles are easy to recognize because they always meet at a straight line segment that divides the space evenly.

Another way to understand it is by looking at a straight angle on a piece of paper, like the radius of a circle, which always measures 180 degrees. Consequently, any pair of adjacent angles within that 180-degree range must add up to 180.

2. Complementary Angles

Complementary angles are closely related to supplementary angles, but instead of summing to 180 degrees, they add up to 90 degrees. These adjacent angles fit within a "square" angle of 90 degrees (or .5π radians).

Complementary angles can be a little more challenging to identify at a glance, as angles like 92 degrees or 88 degrees may still appear to form a square shape to the naked eye.

In most geometry problems, your instructor or textbook will provide a marker to indicate when an angle measures exactly 90 degrees. Typically, this is shown using a small square symbol (instead of a curved arc) inside the angle, commonly seen in textbooks and engineering documents.

3. Vertical Angles

Vertical angles are essentially two pairs of supplementary angles that are adjacent to one another, and when combined, they sum to a complete 360 degrees (or 2π radians).

To recognize vertical angles, you can try drawing a full circle around them using a compass. If the circle touches every segment, you are likely dealing with vertical angles.

The simplest vertical angles occur when two lines intersect, creating four distinct linear pairs of angles that together total 360 degrees.

You can add multiple angles within these segments, which may complicate the calculations. Nonetheless, the total will always equal 360 degrees, regardless of how many adjacent angles are involved.

Other Examples of Adjacent Angles

Supplementary, complementary, and vertical angles are fundamental in geometry because they can be easily recognized visually by both students and professionals.

However, adjacent angles do not necessarily have to fall into these three categories. Some adjacent angle combinations may total values other than 90, 180, or 360 degrees.

Typically, when calculating other types of adjacent angles, your instructor or textbook will provide the total sum, which you can then use to find any unknown angles. Otherwise, you will need to use a tool like a protractor to determine the precise measurement.

How to Calculate Adjacent Angles

The key piece of information when calculating adjacent angles is their total sum, whether that is 90, 180, 360 degrees, or another value. Your teacher may provide this, or you will need to figure it out visually using the rules outlined earlier.

After determining the sum of all adjacent angles, look for any other known angle values in the diagram. You can subtract these from the total sum, and the remaining unknown angles will need to add up to the remainder of the sum (minus any known values).

Example 1

For example, let's say there are three supplementary angles that together total 180 degrees, with two known and one unknown. If the known angles are 30 and 50 degrees, then the third angle will be 180 – (30 + 50), which equals 100 degrees.

Example 2: Combining Opposite Angles and Adjacent Angles

When dealing with more complex adjacent angle problems, it helps to identify opposite angles that are formed by the same line segments.

For instance, imagine six adjacent angles adding up to 360 degrees, making them vertical angles. These angles can be grouped into three pairs of opposite angles. Two values are already known, and four are unknown.

Since opposite angles are always equal, the two angles opposite the known ones can be identified right away. This leaves two unknown angles, which must also be equal to each other:

We can then divide this result by two, giving us 38 degrees for each of the remaining angles.

Another approach to solving this problem is recognizing that vertical angles can be broken down into two sets of supplementary angles:

Opposite Angles

Opposite angles are a distinct concept from adjacent angles but still play a significant role in geometry. They often arise when adjacent angles are drawn, especially in the case of vertical angles.

Your teacher will likely introduce opposite and adjacent angles together since recognizing one can help you determine the other.

When two straight lines cross, the angles formed directly opposite to each other are always equal. Understanding this can provide a quick method for calculating unknown adjacent angles.