Congruent
Geometry & MeasurementTwo 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. You could pick one up and place it on top of the other and they would match perfectly. You can flip, turn, or slide one shape to match the other.
Example
Two triangles are congruent if all three sides and all three angles are equal. Cut two identical pieces of paper in the same triangle shape: they are congruent, even if one is flipped over.
Key Insight
The symbol for congruent is the squiggly equals sign with a regular equals sign underneath (like an = with a ~ on top). Congruent shapes have equal perimeters and equal areas, but equal perimeter/area alone does not guarantee congruence.
Definition
Two figures are congruent if one can be obtained from the other by a sequence of rigid motions (translations, rotations, reflections). Congruent figures have equal corresponding side lengths and equal corresponding angle measures. For triangles, the congruence shortcuts are SSS, SAS, ASA, AAS, and HL.
Example
Triangle with sides $3$, $4$, $5$ and triangle with sides $5$, $4$, $3$ are congruent (SSS). If you rotate or reflect a shape, you still get a congruent figure. Scaling a shape creates a similar figure, not a congruent one.
Key Insight
The triangle congruence shortcuts (SSS, SAS, etc.) are useful because you do not always need all six measurements ($3$ sides, $3$ angles) to confirm congruence. Knowing just $3$ correctly chosen measurements is often enough.
Definition
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. Congruence is an equivalence relation on geometric figures, and congruence classes are the orbits under the group of isometries (rigid motions) $\text{Isom}(\mathbb{R}^2)$.
Example
Two triangles are congruent iff their sorted side-length triples are equal (by SSS). In coordinates, triangles with vertices $\{A,B,C\}$ and $\{A',B',C'\}$ are congruent iff there exists an isometry $T$ with $T(A)=A'$, $T(B)=B'$, $T(C)=C'$, which holds iff the six pairwise distances are preserved.
Key Insight
The group of isometries of $\mathbb{R}^2$ is the Euclidean group $E(2)$, generated by translations, rotations, and reflections. Classifying geometric figures up to congruence is equivalent to studying orbits of $E(2)$, a central problem in the representation theory of symmetry groups.