The direct answer: “Congruent” means two figures have the same shape and the same size; they can be matched exactly by rigid motions (translations, rotations, reflections). The symbol for congruence is $\cong$.
Explanation
- Definition: Two geometric objects are congruent if one can be moved (without stretching or shrinking) to coincide exactly with the other. Rigid motions that preserve distance and angle measure are the allowed moves.
- Symbol/notation:
- Segments: $AB \cong CD$ means segment $AB$ has the same length as segment $CD$.
- Angles: $\angle A \cong \angle B$ means the two angles have equal measure.
- Triangles/figures: $\triangle ABC \cong \triangle DEF$ means triangle $ABC$ is congruent to triangle $DEF$; the vertex order indicates corresponding vertices: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$.
- Correspondence matters: When writing $\triangle ABC \cong \triangle DEF$, the order shows which parts correspond. From this, you can conclude $AB \cong DE$, $BC \cong EF$, $AC \cong DF$, and $\angle A \cong \angle D$, etc. (Often used as CPCTC — corresponding parts of congruent triangles are congruent.)
Common congruence criteria for triangles
To prove two triangles are congruent, you can use any of these conditions (they guarantee exact same shape and size):
- SSS (Side–Side–Side): three pairs of corresponding sides equal.
Example: if $AB = DE$, $BC = EF$, $CA = FD$ then $\triangle ABC \cong \triangle DEF$.
- SAS (Side–Angle–Side): two sides and the included angle equal.
- ASA (Angle–Side–Angle): two angles and the included side equal.
- AAS (Angle–Angle–Side): two angles and a non-included side equal.
- RHS (Right angle–Hypotenuse–Side) for right triangles: hypotenuse and one leg equal.
Rigid motions (how congruence is realized)
- Translation: slide
- Rotation: turn
- Reflection: flip
Any combination of these maps a figure to a congruent figure because they preserve distances and angles.
Congruent vs Similar
- Congruent: same size and shape (exact match via rigid motion).
- Similar: same shape but possibly different size (can be obtained by scaling and rigid motions). Example: a $2\times3$ rectangle and a $4\times6$ rectangle are similar but not congruent.
Quick examples
- Triangles with side lengths $(3,4,5)$ and $(5,3,4)$ are congruent by SSS (reorder sides to match).
- A $3\times5$ rectangle and a $5\times3$ rectangle are congruent because you can rotate one by $90^\circ$.
If you want, I can:
- Give a step-by-step congruence proof for a specific pair of triangles, or
- Provide practice problems with answers (e.g., identify which criterion applies).