Adjacent Angles: Types and Examples

Adjacent angles are one of the earliest and most important concepts to learn in basic geometry, as they have applications in further subjects like trigonometry, physics, and engineering. Effective students of geometry should be able to identify adjacent angles on sight and calculate them to a great degree of accuracy without a protractor.

How to Identify Adjacent Angles

In the simplest terms, these things called adjacent angles are two angles which share a common vertex or line segment and don't overlap with each other.

Knowing the value of one or more adjacent angles can be helpful in measuring the value of unknown angles, depending on the various properties of adjacent angles and the angles as a group. If you know the sum of all angles with common vertex, for instance, this will make calculating adjacent angles much easier.

While the simplest adjacent angles are made up of only two angle values, many pairs of adjacent angles may be adjacent to each other, creating values and angle relationships which may add up to sums as high as (but never exceeding) 360 degrees.

3 Types of Adjacent Angles

1. Supplementary Angles

Supplementary angles are adjacent angles which add up to 180 degrees (or π radians) when combined. Supplementary angles are simple to identify visually because because they will all intersect at a line segment which cuts straight across.

Another way to think of it is that a perfectly straight angle on a piece of paper, such as the radius of a circle, will always have a value of 180 degrees. Therefore, any adjacent angles contained within that 180 degree sweep will necessarily have to add up to 180.

2. Complementary Angles

Complementary angles are very similar to supplementary angles but add up to 90 degrees instead of 180. As such, these adjacent angles must be contained within a "square" angle of 90 degrees (.5π radians).

Complementary angles can be a bit trickier to identify visually because values like 92 degrees or 88 degrees may still look square to the naked eye.

Generally, while doing geometry problems, your instructor or textbook writer will include an indicator to show when an angle is exactly 90 degrees. In many textbooks, as well as engineering documents, 90-degree corners are called out using a small square icon (rather than a curved arc) inside the angle.

3. Vertical Angles

Vertical angles can be thought of as two sets of supplementary angles which are considered adjacent angles to each other. Therefore, they add up to a full 360 degrees (2π radians).

To identify vertical angles visually, you can try drawing a full circle around them with a compass. If the circle is able to connect with every segment, you're probably dealing with vertical angles.

The simplest vertical angles are just two lines which intersect straight through each other, creating four linear pairs of discrete angles which add up to 360.

Any number of angles can be added inside these segments, complicating the calculation process. However, the sum will always be 360 no matter how many adjacent angles are involved.

Other Adjacent Angles Examples

Supplementary angles, complementary angles and vertical angles are all useful in studying geometry because they can be identified visually by the student or worker.

However, there is no rule stating that adjacent angles have to fit into those three categories. These examples of adjacent angles will add up to values other than 90, 180 or 360 degrees.

Generally, if you are required to calculate other types of adjacent angles, your instructor or textbook will tell you the total sum so that you can use that value to calculate any unknown angles. Otherwise, you'll need a manual tool such as a protractor to measure the exact value.

Calculating Adjacent Angles

The most important piece of information to know when calculating adjacent angles is their sum, whether this value is 90, 180, 360 or some other number of degrees. Your instructor may provide you with this information, or you will have to identify it visually using the rules stated in previous sections.

Once you know the sum of all adjacent angles, scan the drawing for any other angle values which are already known. These can be subtracted from your sum, and now you know that the total of any unknown values will have to add up to that sum (minus any known values).

Example 1

For instance, let's say we have three supplementary angles which add up to 180 degrees, two of which are known, and one unknown. If the two known adjacent angle values are 30 and 50 degrees, then our third angle measurement will have to be 180 – (30 + 50), or 100 degrees.

Example 2: Combining Opposite Angles and Adjacent Angles

When it comes to calculating more complex adjacent angle problems, it will be helpful to identify opposite angles formed by the same line segments.

For example, let's say six adjacent angles add up to 360 degrees making them vertical angles, and they can be sorted into three pairs of opposite angles. Two values are known, and four are unknown.

Since we know that opposite angles have to be equal, we can identify the values of the two angles opposite to our known values immediately. This leaves only two unknown angles left to calculate, which will also be equal to each other:

We can then divide that calculation by two to arrive at a value of 38 degrees for each remaining angle.

Another way to solve this problem is to understand that vertical angles can be split into two sets of supplementary angles:

Opposite Angles

Opposite angles are a separate concept to adjacent angles but similarly important in geometry. Opposite angles may also appear when adjacent angles are drawn, especially in the case of vertical angles.

Your instructor will likely teach you about opposites and adjacent angles together because identifying one can help you calculate the other.

When two straight lines intersect, the angles formed which are directly opposed to one another always have an equal value. Knowing this fact can give you an easy shortcut when it comes to calculating unknown adjacent angle measurements.

Original article: Adjacent Angles: Types and Examples

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