Triangle Similarity Test - Two sides and included angle (SAS)

 

Definition: Triangles are similar if two sides in one triangle are in the same proportion to the corresponding sides in the other, and the included angle are equal.

This (SAS) 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 two sides in one triangle are in the same proportion to the corresponding sides in the other, and the included angle between them is the same then the triangles are similar.

For example in the triangle above, the side PQ is exactly twice as long as the corresponding side LM in the other triangle, and PR is twice LN. So these two sides are in the same proportion, in this case 2:1 (two to one). Also the included angle angles (P and L) are equal in measure, and so the triangles are similar.

Notice that the two sides and the angle are drawn in a magenta color to show they are the things being used to test for similarity.

What does this mean?

Since two corresponding pairs of sides are in the same proportion, and the included angle angles are equal, we can be sure the triangles are similar.

Because the triangles are similar:

  1. The three angles at P,Q and R are equal to the angles L,M and N respectively.
  2. The corresponding sides in each triangle will all be in the same proportion.

But don't forget

Two triangles, similar because all three sides are the same length, but one 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, the two triangles are still similar, even though one is the mirror image of the other and rotated.

Related topics

Similar Polygons

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