High School: Geometry
7. Use the definitions of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Students should know that two triangles are congruent if there is a rigid motion that maps one onto the other. All corresponding pairs of all sides and all angles must be congruent, which means the two triangles are essentially equal. The only triangles that don't work like this are love triangles; each one is different in its own heartbreaking way.
Of course, if you've read through the standards before this, you know that rigid motions and congruence aren't limited to triangles. So why do triangles get their own standard? What makes them so special?
Since triangles are defined by 3 sides and 3 angles, knowing a limited number of each is often enough to find the rest of the missing information. In short, congruence is easier to prove with triangles. In fact, triangles have their own postulates that are designed to prove triangle congruence with limited information specifically. Fancy shmancy.
So if all corresponding sides and angles of two triangles are congruent, the two triangles themselves are congruent. In applying the rules of congruence and rigid motions to triangles, students drill down one level deeper: not only are the triangles themselves congruent, but the corresponding parts are congruent, as well. This opens up a whole new world of how to discover more about the use of triangles in the real world.