Noncongruent

Noncongruent is a term used in geometry to describe figures that are not congruent to each other. Two figures are congruent if one can be transformed into the other by a rigid motion in the plane, meaning a combination of translation, rotation, and reflection. Therefore, noncongruent figures cannot be matched by any such transformation; they differ in size, shape, or both.

Common examples help illustrate the concept. Two squares with different side lengths are noncongruent. Two triangles with different sets of side lengths, such as 3-4-5 and 2-3-4, are noncongruent. Two rectangles that have the same area but different side ratios can also be noncongruent. By contrast, copies of the same triangle placed in different positions are congruent.

It is important to distinguish noncongruence from similarity. Two shapes can be similar (same shape, different size) and yet be noncongruent if their sizes differ. Conversely, congruent shapes are identical in size and shape, regardless of orientation.

Determining noncongruence involves checking whether any rigid motion can map one figure to the other. For polygons, this typically requires comparing an ordered sequence of side lengths and interior angles; a mismatch under every possible vertex correspondence proves noncongruence. In particular, having equal sets of side lengths is not sufficient for congruence unless there is a consistent correspondence of sides and angles.

In practice, the concept helps classify figures and count distinct forms under rigid motion, such as enumerating noncongruent triangles with given constraints or identifying noncongruent polygons within a family. In circles, two circles are congruent if and only if their radii are equal; circles with different radii are noncongruent.