In elementary geometry the word congruent is often used as follows. Note hatch marks are used here to show angle and side equalities. This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'.
So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely.
This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. The unchanged properties are called invariants. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The last triangle is neither congruent nor similar to any of the others. The two triangles on the left are congruent, while the third is similar to them.
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