Congruence
Definition
A geometric figure \(\Omega_{1} \subseteq \mathbb{R}^{n}\) is said to be congruent to \(\Omega_{2} \subseteq \mathbb{R}^{n}\), denoted by \(\Omega_{1} \cong \Omega_{2}\), if there is as isometry \(\tau\) such that \(\tau(\Omega_{1}) = \Omega_{2}\).
Theorem
Congruence is an equivalence relation.
This fact is very closely related to the fact that isometries form a group under composition.
Proof
Reflexivity is trivially true because the identity function is an isometry.
Symmetry follows from the fact that isometries are invertible, with the inverse an isometry as well, and hence:
\[\tau(\Omega_{1}) = \Omega_{2} \implies \tau^{-1}(\Omega_{2}) = \Omega_{1}.\]
Transitivity follows from the fact that isometries compose to form other isometries, and hence
\[\tau(\Omega_{1}) = \Omega_{2} \quad \text{and} \quad \alpha(\Omega_{2}) = \Omega_{3} \implies (\alpha \circ \tau)(\Omega_{1}) = \Omega_{3}.\]