Research-based guidance and classroom activities for teachers of mathematics

Similarity and congruence

Similarity and congruence is a theme within spatial and geometrical reasoning. Links to relevant activities and resources are on the right hand side of this page. 

Similarity is an important idea that relates enlargement, scale factor, area growth, indirect measurement, and projective geometry.

Mathematically similar geometric shapes (ones that have the same shape but may differ in size) provide helpful mental images of ratios and equivalent fractions. Ideas of similarity extend to trigonometry and to the notion of self-similarity that is characteristic of fractal geometry.

Two geometric figures are said to be similar when all corresponding angles are equal and all corresponding distances are in the same ratio. The figures are congruent if they have precisely the same shape and size, or if one has the same shape and size as the mirror image of the other.

A similarity can also be defined as a transformation that preserves ratios of distances, while two geometric figures are said to exhibit geometric congruence (or be geometrically congruent) if (and only if), one can be transformed into the other by an isometry. The use of transformations can be a means by which ideas of invariance can be studied most easily, and by which the formal definitions of similarity and congruence can be related to learners’ previous intuitive ideas. Dynamic geometry software (DGS) can play a valuable role in this.

Activities and resources