Congruence

Geometric Shapes

Definition: Geometric Figure

A geometric figure in the Euclidean space ${\mathbb{R}}^n$ is any subset $G\subset {\mathbb{R}}^n$.

Similarity

Definition: Similarity

Two geometric shapes $G_1$ and $G_2$ are similar iff there exists a transformation $f$ which is a combination of a translation and a scaling such that

$$f(G_1)=G_2\text{ and }{f}^{-1}(G_2)=G_1$$

NOTATION

$$G_1\sim G_2$$

Congruence

Definition: Congruence

Two geometric shapes $G_1$ and $G_2$ are congruent if there exists an isometry which maps $G_1$ onto $G_2$.

NOTATION

$$G_1\cong G_2$$

INTUITION

The two geometric figures are congruent if there is a way to move and rotate one of them in such a way so as to make it coincide with the other figure.

THEOREM

Congruence is an equivalence relation.

PROOF

TODO

Bibliography