nLab congruence in geometry

Two triangles in Eucledean plane are said to be congruent if their corresponding sides are equal, $a' = a$, $b' = b$, $c' = c$, and the corresponding angles are also equal, $\alpha'=\alpha$, $\beta' = \beta$ i $\gamma' = \gamma$.

It appears that two triangles are congruent iff there is an isometry of the plane which sends one onto another bijectively (for this it is sufficient to send the vertices to vertices, as every isometry is a bijection and it sends convex hulls onto convex hulls). Such an isometry is unique because if there are two, say $g$ and $h$ then $h^{-1}\circ g$ is an isometry fixing the first triangle and the only isometry of the plane fixing three noncolinear points is the identity.

This motivates how to define congruence of more complicated figures: two plane figures are congruent if there is an isometry sending one onto another bijectively; similarly for angles. For oriented angles and oriented figures we need to consider just isometries preserving orientation. In the plane these are isometries which can be represented as a composition of an even number of reflections.

Related pages: axioms of plane geometry

category: geometry