Congruence (geometry and related uses)
Congruence describes when two figures have identical shape and size. In geometry it is formalized by isometries; applied criteria exist for polygons and triangles and contrasts with similarity and modular congruence.
In elementary geometry, two figures are called congruent when one can be moved so that it coincides exactly with the other without changing size. Informally, congruent shapes have the same shape and size; the relation is commonly written with the symbol ≅. A simple physical test is to cut two plane figures from paper and see whether one can be placed over the other to match perfectly — turning the paper over (reflection) is allowed.
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Formal definition and transformations
More precisely, two sets of points in Euclidean space are congruent if there exists an isometry that carries one set onto the other. An isometry is a distance-preserving map and can be realized by a composition of rigid motions: translations, rotations and reflections (and in the plane, glide reflections). These motions do not alter distances or angles, so lengths and angles are invariant under congruence.
Key properties and common criteria
Congruence is an equivalence relation: it is reflexive, symmetric and transitive. For practical work with polygons and especially triangles, several standard tests determine congruence without constructing an explicit isometry. For triangles the most used criteria are:
- SSS (side-side-side): all three pairs of corresponding sides equal;
- SAS (side-angle-side): two sides and the included angle equal;
- ASA (angle-side-angle): two angles and the included side equal;
- RHS or HL (right-angle, hypotenuse, side): for right triangles.
These tests rely on the fact that rigid motions are determined by matching lengths and included angles so that one triangle can be superimposed on the other exactly.
Distinctions, uses and related meanings
Congruence differs from similarity: similar figures have the same shape but may differ in size because similarity allows uniform scaling. Congruence forbids scaling. The term also appears outside geometry: in number theory, a different notion called congruence modulo n relates integers that leave the same remainder on division by n. Within geometry, congruence is central to construction problems, proofs, computer graphics (rigid-body transformations), pattern matching and manufacturing where exact size preservation matters.
Historical notes and further reading
The concept of congruence has roots in ancient Greek geometry where congruent figures were compared by superposition. Modern formalization uses group theory of isometries and metric properties. For basic introductions and formal treatments see texts on geometry, sets of points and transformations; for precise set-theoretic definitions consult resources on sets of points and metric geometry. For elementary discussions of shape see resources on shape and for polygon-specific results consult material about polygons.
Congruence remains a foundational idea that links visual intuition (matching shapes) with rigorous mathematical structure (isometries and equivalence relations), and underpins many practical tasks from drafting to computer vision.
Questions and answers
Q: What does it mean for two figures to be congruent in geometry?
A: Two figures are congruent in geometry if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
Q: How are two sets of points called congruent?
A: Two sets of points are called congruent if and only if one can be transformed into the other by isometry.
Q: What are rigid motions used for in isometry?
A: Rigid motions are used in isometry to reposition, rotate or reflect geometrical figures without resizing them, so that they coincide exactly with other objects.
Q: Can two figures be congruent if one of them has to change its size to coincide with the other?
A: No, if one of the objects has to change its size to coincide with the other, then the two objects are not congruent, but they are called similar.
Q: What can we say about the congruence of two distinct plane figures on a piece of paper?
A: Two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely, turning the paper over if needed.
Q: What are congruent polygons?
A: Congruent polygons are polygons that can be folded in half to form another regular polygon that is also congruent.
Q: What is the criterion for two objects to be called congruent in geometry?
A: The criterion for two objects to be called congruent in geometry is that one object can be repositioned, rotated or reflected so that it coincides exactly with the other object, without changing its size.
Related articles
Author
AlegsaOnline.com Congruence (geometry and related uses) Leandro Alegsa
URL: https://en.alegsaonline.com/art/22523
Sources
- web.cortland.edu : "Oxford Concise Dictionary of Mathematics, Congruent Figures"