Reflex Angle: Definition & Examples

Reflex Angle: Definition & Examples
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  • 0:00 What Is A Reflex Angle?
  • 0:35 Acute And Obtuse vs. Reflex
  • 1:25 How To Measure A Reflex Angle
  • 2:55 Example 1
  • 3:50 Example 2
  • 5:05 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Reflex angles are used less frequently than other angles, but are easily identified in our everyday environment. This lesson defines reflex angles, explains how to measure them, and examines the relationship between reflex, acute, and obtuse angles.

What Is A Reflex Angle?

You have probably heard of acute and obtuse angles. An acute angle is an angle that measures between 0 and 90 degrees, and an obtuse angle is an angle that measures between 90 and 180 degrees. There is another type of angle that we hear much less about called a reflex angle. Reflex angles are angles measuring greater than 180 degrees and less than 360 degrees. The measure of a reflex angle is added to an acute or obtuse angle to make a full 360 degree circle.

Figure 1: Note how each angle is labeled with its degree of measure.
reflex, acute, and obtuse angle

Acute and Obtuse vs. Reflex

We can observe reflex angles in our everyday environment. Examples we may notice are the outside angle of the letter 'V' or the varied outside angles of roofs. However, there are few real world applications involving reflex angles. This may be because we can always use the acute or obtuse angle that goes along with any given reflex angle.

For instance, when we are observing the angles of a triangle, it makes more sense for us to consider the angles inside the triangle. Even though each of these angles has a reflex angle on the outside, we are much more inclined to work with the acute or obtuse angles on the interior of the triangle. See figure 2.

Figure 2: Reflex Angles
triangle reflex angles

One more reason we work with acute and obtuse angles more frequently than we work with reflex angles is that the measure of acute and obtuse angles is less than the measure of reflex angles. Therefore, acute and obtuse angles can be easier to work with.

How to Measure a Reflex Angle

Although reflex angles rarely show up in real world applications, there may be an instance when you need to measure an angle that is greater than 180 degrees. As a result, it is useful to know how to do this. When we want to measure an acute or obtuse angle we can easily use a protractor to do so. However, a protractor only measures angles up to 180 degrees. So how can we measure a reflex angle?

First let's recap what is known. A reflex angle's measure is added to an acute or obtuse angle to make a full 360 degree circle.

Figure 3: Note the measure of the reflex angle in each example.
reflex angle relationship

This means if we have an acute or obtuse angle measuring x degrees, then there is a reflex angle measuring r degrees such that x + r = 360. If we subtract x from both sides of this equation, we have r = 360 - x. This gives us a formula to find the measure of a reflex angle when the corresponding acute or obtuse angle is known.

If the corresponding acute or obtuse angle is unknown, then use a protractor to first determine its measure. Subtract the result from 360 degrees. This difference is the measure of the reflex angle.

Consider the first reflex angle shown in figure 3. If we use a protractor to measure the acute angle that goes along with this reflex angle, we see that the acute angle has a measure of 55 degrees. Therefore, our reflex angle has a measure of 305 degrees (360-55=305).

Example 1

Let's look at an example. Suppose we have a triangle with interior angles 115, 42, and 23 degrees. What are the reflex angles (outside angles) of the triangle?

Solution:

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