Two figures are congruent if there is a sequence of translations, rotations, and reflections that takes one figure onto the other. This is because translations, rotations, and reflections are rigid motions. Any sequence of rigid motions is called a rigid transformation. A rigid transformation is a transformation that doesn’t change measurements on any figure. With a rigid transformation, figures like polygons have corresponding sides of the same length and corresponding angles of the same measure. The fact that rigid transformations always take lines to lines, angles to angles of the same measure, and segments to segments of the same length seems to be true, but there is no way to prove or disprove this. This means rigid transformations are an assertion—an observation that seems to be true, but is not proven.

The result of any transformation is called the image. The points in the original figure are the inputs for the transformation sequence and are named with capital letters. The points in the image are the outputs and are named with capital letters and an apostrophe, which is referred to as “prime.”

Each step in this sequence of rigid transformations creates a triangle that is congruent to triangle .

Two images of triangle A B C  in a transformation sequence. First image shows triangle A B C translated downward and left resulting in triangle A prime B prime C prime, then is translated right to resulting in triangle A double prime B double prime C double prime. Second image shows triangle A B C translated right resulting in triangle A prime B prime C prime, then is translated downward and left resulting in triangle A double prime, B double prime, C double prime.