If all pairs of corresponding sides and angles in two triangles are congruent, then it is possible to find a rigid transformation that takes corresponding vertices onto one another. This proves that if two triangles have all pairs of corresponding sides and angles congruent, then the triangles must be congruent. But justifying that the vertices must line up does not require knowing all the pairs of corresponding sides and angles are congruent. We can justify that the triangles must be congruent if all we know is that two pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. This is called the Side-Angle-Side Triangle Congruence Theorem.

To find out if two triangles, or two parts of triangles, are congruent, see if the given information or the diagram indicates that 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. If that is the case, we don’t need to show and justify all the transformations that take one triangle onto the other triangle. Instead, we can explain how we know the pairs of corresponding sides and angles are congruent and say that the two triangles must be congruent because of the Side-Angle-Side Triangle Congruence Theorem.

Sometimes, to find congruent triangles, we may need to add more lines to the diagram. We can decide what properties those lines have based on how we construct the lines (An angle bisector? A perpendicular bisector? A line connecting two given points?). Mathematicians call these additional lines auxiliary lines because auxiliary means “providing additional help or support.” These are lines that give us extra help in seeing hidden triangle structures.