Similar Triangles & the AA Criterion

Similar Triangles & the AA Criterion
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  • 0:01 The AA Criterion
  • 1:28 Similar Triangles
  • 2:27 Why It Works
  • 3:08 Example
  • 4:09 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

This video lesson explains the AA criterion and shows you why you only need two corresponding angles to be equal for two triangles to be similar. You will also see how you can use it to help you identify similar triangles.

The AA Criterion

When you have two triangles where one is a smaller version of the other, you are looking at two similar triangles. The mathematical definition for similar triangles states that the triangles have proportional corresponding sides and all the corresponding angles are the same. One triangle will look like the miniature of the other. When we say that the corresponding sides are proportional, it just means that all sides of one triangle are the same amount larger or smaller than the other triangle.

For example, these two similar triangles show that each side of the larger triangle is twice as large as the smaller triangle. We don't have one side of the larger triangle being twice as big as the smaller triangle's corresponding side and another side being three times as big. No, all the sides are the same amount larger or smaller.

similar triangles

And of course, all the corresponding angles are equal to each. This means that if the top angle measures 60 degrees in the larger triangle, then the top angle of the smaller triangle also equals 60 degrees.

In math, we have what is called the AA criterion. This criterion tells us that two triangles are similar if two corresponding angles are equal to each other. That's right, all you need are two corresponding angles to be similar for your triangles to be similar.

Similar Triangles

Why is this? Well, think about the properties of the angles of a triangle. What do they all add up to for every triangle? They add up to 180 degrees. Yes, all triangles will have a total of 180 degrees from the three angles. Because we know this total, if we know the measurements of two of the angles, we automatically know the measurement of the third.

For example, if two angles of a particular triangle measure 45 degrees and 45 degrees, then the third angle will measure 90 degrees because 180 - 45 - 45 = 90. If another triangle also has two angles that both measure 45 degrees, then that triangle will also have a third angle that measures 90 degrees. So if two corresponding angles of two triangles are equal, then the third corresponding angle will also be equal. This is why you only need to look at two angles.

Why It Works

This criterion works because when you make a triangle smaller or larger, you don't move the location of the sides. The only thing that changes is the length of the sides; the angles always stay the same. You can see this for yourself by measuring the angles of similar triangles. Try measuring the angles of these two triangles:

similar triangles

You'll see that all the corresponding angles are the same. Also, if you measure just two of the angles, the third one can be found by subtracting those two angles from 180. So if two corresponding angles are the same, then the third angle will also be the same since the same number is being subtracted from 180.

Example

Let's see how we can use this AA criterion to help us identify similar triangles.

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