Which congruence proof establishes similarity by showing that two sides are equal and the angle between them matches?

The congruence proof that establishes similarity by showing that two sides are equal and the angle between them matches is known as the Side-Angle-Side (SAS) theorem. This theorem states that if two sides of a triangle are proportional to two sides of another triangle and the angle included between those sides is equal, then the two triangles are similar.

In detail, similarity in triangles is determined by the ratio of corresponding sides and the angles between those sides. When two sides are equal and the angle included is the same, it guarantees that the third side must also be in proportion, leading to the conclusion that the triangles are similar.

This approach contrasts with other methods like Angle-Angle-Side (AAS), where similarity is established using angles rather than sides. The Side-Side-Side (SSS) criterion requires all three sides to be proportional, and the Right angle-Hypotenuse-Side (RHS) applies specifically to right triangles, focusing on a combination of a right angle and the lengths of hypotenuse and one other side. Thus, SAS is the most direct method for establishing the criteria presented in the question.