CONGRUENCE, a mathematical term employed in several senses, each of them connoting harmonious relation, agreement or correspondence.
Two geometric figures are said to be congruent, or to be in the relation of congruence, if it is possible to superpose one of them on the other so that they shall coincide throughout. Thus two triangles are congruent if two sides and their included angle in the one are equal to two sides and their included angle in the other. This idea of congruence seems to be founded on that of a "rigid body," which may be moved from place to place without change in the internal relations of its parts. But it must rest on a previous concept of metrical relations among the parts of the body, since otherwise there would be no basis on which to determine whether the body had changed in shape and size.
The position of a (straight) line (of infinite extent) in space may be specified by assigning four suitably chosen co-ordinates. A congruence of lines in space is the set of lines obtained when the four co-ordinates of each line satisfy two given conditions. For ex ample, all the lines cutting each of two given curves form a con gruence. The co-ordinates of a line in a congruence may be ex pressed as functions of two independent parameters; from this it follows that the theory of congruences is analogous to that of surfaces in space of three dimensions. An important problem for a given congruence is that of determining the simplest surface into which it may be transformed.