Congruence (geometry)

Congruence is the relation between geometric figures that coincide in shape and size under rigid motions. Congruent figures have equal corresponding lengths and angles.

Rigid motion: A transformation of space that preserves distances and angles. Congruent figures are precisely those related by distance-preserving transformations.
Side-side-side: A triangle congruence criterion requiring equality of all three corresponding side lengths. Three matching side lengths determine a triangle up to rigid motion.
Similarity (geometry): A relation between figures with equal angles and proportional corresponding lengths. Similar figures may differ in size, whereas congruent figures must match in size.
Triangle: A polygon with three sides and three angles. Triangle congruence criteria are among the most-used tools for proving geometric facts.
Isometry: A distance-preserving map between metric spaces. Isometries formalize the transformations that carry one congruent figure onto another.
Side-angle-side: A triangle congruence criterion using two sides and their included angle. The included angle fixes how the two known sides meet.
Affine transformation: A transformation preserving collinearity and ratios along parallel lines, but not generally lengths or angles. Affine maps can stretch a figure, so they do not generally preserve congruence.
Geometric proof: A deductive argument establishing a statement about geometric objects. Congruence transfers equal sides and angles between figures within a proof.
Euclidean distance: The ordinary straight-line distance between points in Euclidean space. Preservation of Euclidean distances ensures corresponding points remain equally far apart.
Angle-side-angle: A triangle congruence criterion using two angles and their included side. The side and adjacent angles uniquely determine the triangle's shape and scale.

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