<P> 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 . This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object . So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely . Turning the paper over is permitted . </P> <P> In elementary geometry the word congruent is often used as follows . The word equal is often used in place of congruent for these objects . </P> <Ul> <Li> Two line segments are congruent if they have the same length . </Li> <Li> Two angles are congruent if they have the same measure . </Li> <Li> Two circles are congruent if they have the same diameter . </Li> </Ul> <Li> Two line segments are congruent if they have the same length . </Li>

The conditions which are not sufficient for congruent triangles are