Congruent triangles

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In Euclidean geometry, two triangles are congruent if there is a rigid motion which brings one triangle exactly onto the other ("superposition"). Since properties of Euclidean geometry are determined by the Euclidean distance, which in turn determines angles, two triangles are congruent when their configuration is described by the same set of distances and angles. It is a matter of convention whether triangles are regarded as congruent if the motion that transforms one into the other is orientation-reversing, such as a reflection, but this is usually the case.

Criteria for congruence

There are a number of traditional criteria for congruence in terms of measures of corresponding sides or angles. We may regard these as stating that the data are sufficient to construct the triangle unambiguously.

  • SSS (side-side-side): the lengths of corresponding sides are equal in the two triangles;
  • ASA (angle-side-angle): two angles and the side they have in common are equal;
  • SAS (side-angle-side): two sides and the enclosed angle are equal;
  • RHS (right angle-hypotenuse-side): the triangles are right-angled and the hypotenuse and another side are equal;

There is a well-known fallacious criterion, ASS, when two sides and an angle not enclosed are equal. In general there are two non-congruent triangles corresponding to this data, unless the angle is a right angle, when we have the valid RHS criterion.

We may also mention that AAA is a criterion for similarity of triangles, but not congruence, as it fails to prescribe any scale factor.