Supplementary Angle: Definition & Theorem

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  • 0:05 Review of Angles
  • 0:46 Properties
  • 1:49 Theorems
  • 3:01 Applications
  • 3:53 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Often

Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

There are many important angle pairs in geometry. In this lesson, we'll learn about supplementary angles, including what they are, where they are used, and why you need to know about them.

Review of Angles

I work in a technical high school, and many times I hear students ask, 'Why is all this math important? I can understand learning fractions, but why all this stuff about angles?' Well, it turns out that many different types of angles are used in skilled trades, and being able to solve problems involving angles, especially supplementary angles, is a valuable skill to have.

In order to understand the material in this lesson, we'll want to review the different types of angles.

  • Acute angle - measure is between 0 and 90 degrees
  • Right angle - measure is exactly 90 degrees
  • Obtuse angle - measure is between 90 degrees and 180 degrees
  • Straight angle - measure is exactly 180 degrees

Properties of Supplementary Angles

In geometry, two angles whose measures sum to 180 degrees are supplementary angles. These angles may share a common side, a common vertex, or have no points in common.

SupplementaryAngles

Let's look at some examples of supplementary angle pairs.

  • 10 degrees and 170 degrees
  • 30 degrees and 150 degrees
  • 50 degrees and 130 degrees
  • 70 degrees and 110 degrees
  • 90 degrees and 90 degrees

Perhaps you noticed a pattern in this list - except for one pair of two right angles, all the supplementary angle pairs had one acute angle and one obtuse angle. This is an important property of supplementary angles - you will either have two right angles in the supplementary pair, or one acute angle and one obtuse angle.

Remember, only a pair of angles can be supplementary. Sure, the three angles in a triangle may add up to 180 degrees, but there are three angles in a triangle, so they are not supplementary!

Theorems Involving Supplementary Angles

There are a number of theorems in geometry that involve supplementary angles. Keep in mind that since they are theorems, you could end up having to prove that they are true when you take a geometry class!

Congruent supplements theorem - This theorem states that if two angles, A and C, are both supplementary to the same angle, angle B, then angle A and angle C are congruent. That is, angle A and angle C have the same measure.

CongruentSupplements

Same side interior angles - When two parallel lines are crossed by a third line, eight angles are formed. The angles that are in between the two parallel lines, and on the same side of the third line, are called same side interior angles and they are supplementary. Here, angles 1 and 2 are an example of same side interior angles.

SameSideInteriorAngles

Consecutive angles in a parallelogram - You can use the previous theorem to prove that any two consecutive angles in a parallelogram are supplementary. In other words, as you travel around a parallelogram, each angle that you encounter will be supplementary to both the previous angle and the next angle. Here, angle A in the parallelogram is 115 degrees and angle D is 65 degrees, for a sum of 180 degrees.

SupplementsParallelogram

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