Congruent
Two figures are congruent if they have the same shape and size, meaning one can be mapped onto the other by a rigid motion.
Definition
Two shapes are congruent if they are exactly the same shape and size, so that one could be picked up (flipped, turned, or slid) and placed exactly on top of the other; equivalently, one figure can be obtained from the other by a sequence of rigid motions (translations, rotations, reflections), and congruent figures have equal corresponding side lengths and angle measures. For triangles, the congruence shortcuts SSS, SAS, ASA, AAS, and HL confirm congruence without needing all six measurements. Formally, two figures in $\mathbb{R}^2$ are congruent if there exists an isometry (distance-preserving map) sending one to the other; every isometry of the Euclidean plane is a composition of at most $3$ reflections, and congruence is an equivalence relation whose classes are the orbits under the group of isometries $\text{Isom}(\mathbb{R}^2)$.
Example
Two triangles are congruent if all three sides and all three angles are equal; a triangle with sides $3$, $4$, $5$ and one with sides $5$, $4$, $3$ are congruent by SSS, even if one is flipped over (a mirror image). Rotating or reflecting a shape always produces a congruent figure, while scaling creates a similar figure, not a congruent one. In coordinates, triangles with vertices $\{A,B,C\}$ and $\{A',B',C'\}$ are congruent if and only if there exists an isometry $T$ with $T(A)=A'$, $T(B)=B'$, $T(C)=C'$, which holds exactly when all six pairwise distances are preserved.
Key Insight
The symbol for congruent is a squiggly equals sign over a regular equals sign; congruent shapes always have equal perimeters and equal areas, but equal perimeter or area alone does not guarantee congruence. The group of isometries of $\mathbb{R}^2$ is the Euclidean group $E(2)$, generated by translations, rotations, and reflections, and classifying geometric figures up to congruence is equivalent to studying orbits of $E(2)$, a central problem in the representation theory of symmetry groups.