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Things that match are said to be congruent to one another. Congruent figures are figures that match; they must be the same size and shape; one can be superimposed on the other in such a way that all corresponding parts will match exactly.
In geometry, we generally speak of the geometric objects (segments, angles, triangles, etc.) as being congruent (or not), but the measurements of these objects (lengths, numbers of degrees, area, etc.) as equal (or not). For example, two line segments are congruent if (and only if) their lengths are equal.
We can also speak of numbers being congruent, if we specify what we mean by 'matching.' For example, the numbers 2, 7, 12, 17, … 137, … are "congruent modulo 5," meaning that "they match" (their remainders are the same) when they are divided by 5. Likewise, odd numbers are congruent modulo 2, because all their remainders are the same when divided by 2. Even numbers are, of course, also congruent modulo 2, because when any of them is divided by 2, "there is no remainder" (that is, the remainder is 0).
In geometry, transformations are often classified according to whether they preserve congruence. For example, when a polygon is translated, or slid, the image is congruent to the original polygon. That is, the lengths of its sides and the measures of its angles are that same. Rotations and reflections also preserve congruence. Sometimes, these transformations are called rigid motions.
Related mathematical terms
What's in a word?
From Latin ‘to agree.’
Other related words
incongruous (‘doesn’t fit; doesn't match; doesn't quite agree’)