Besides the Angle-Angle Triangle Similarity Theorem, what other criteria are sufficient to prove triangles similar?

When two sides of one triangle are proportional to two corresponding sides of a second triangle, using the same scale factor, , and the pair of angles between these sides are congruent, then the triangles are similar by the Side-Angle-Side Triangle Similarity Theorem.

For example, angles and are vertical angles and so they are congruent, and there are two pairs of corresponding sides with the same scale factor.

Two triangles, E D F and B D C. Angles E D F and B D C are vertical angles. Point E prime plotted on segment D E. Point F prime plotted on segment F D. Segment E prime F prime drawn. Segment D C labeled k h. Segment D B labeled k j. Segment D E prime labeled k j. Segment D F prime labeled k h.  

Dilate triangle using center and a scale factor of . Because  , is now congruent to , and is congruent to . The dilation did not change the size of the angles. Therefore, triangle is congruent to triangle by the Side-Angle-Side Triangle Congruence Theorem. This means that there is a sequence of rigid motions that takes triangle to triangle . That means that triangle is similar to triangle because there is a dilation and a sequence of rigid motions that takes one to the other. There wasn’t anything special about these two triangles. Therefore, any pair of triangles with two pairs of sides whose lengths are in the same proportion and with the angle between them congruent must be similar.

We can also show that if all three pairs of corresponding sides are proportional and use the same scale factor, , this is sufficient to prove that the triangles are similar. We call this the Side-Side-Side Triangle Similarity Theorem.