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Triangle Similarity Test - All corresponding angles equal (AAA)

 

Definition: Triangles are similar if the measure of all three interior angles in one triangle are the same as the corresponding angles in the other.

This (AAA) is one of the three ways to test that two triangles are similar . For a list see Similar Triangles.

Try this Drag any orange dot at P,Q,R. The triangle LMN will change to remain similar to the left triangle PQR. (If there is no image below, see support page.)

If all three angles in one triangle are the same as the corresponding angles in the other, then the triangles are similar. So for example, in the triangle above the interior angle ∠P is exactly equal to the corresponding angle ∠L in the other triangle. ∠Q is equal to ∠M, and ∠R is equal to ∠N. And so, because all three corresponding angles are equal, the triangles are similar.

Notice that the the three angles are drawn in thick blue lines to indicate they are the parts being used to test for similarity.

What does this mean?

  1. Since all three corresponding angles are equal, we can be sure the triangles are similar.
  2. Because the triangles are similar, this means that the three pairs of corresponding sides are in the same proportion to each other. (See Similar Triangles)

But don't forget

Two triangles, similar because each angle in one is equal to the corresponding angle in the other, but one triangle is flipped and rotated Similar triangles can be rotated and/or mirror images of each other (reflected). (See Similar triangles.) In the figure on the right, each angle in one triangle is equal to the corresponding angle in the other, and so are still similar even though triangle is the mirror image of the other, rotated and bigger.

Related topics

Similar Polygons

(C) 2007 Copyright John Page