Congruence (geometry)

[1] 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.

Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely.

The related concept of similarity applies if the objects have the same shape but do not necessarily have the same size.

Sufficient evidence for congruence between two triangles in Euclidean space can be shown through the following comparisons: The ASA postulate is attributed to Thales of Miletus.

The SSA condition (side-side-angle) which specifies two sides and a non-included angle (also known as ASS, or angle-side-side) does not by itself prove congruence.

In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides.

There are a few possible cases: If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SSA, or long side-short side-angle), then the two triangles are congruent.

[5][6] In more detail, it is a succinct way to say that if triangles ABC and DEF are congruent, that is, with corresponding pairs of angles at vertices A and D; B and E; and C and F, and with corresponding pairs of sides AB and DE; BC and EF; and CA and FD, then the following statements are true: The statement is often used as a justification in elementary geometry proofs when a conclusion of the congruence of parts of two triangles is needed after the congruence of the triangles has been established.

Two conic sections are congruent if their eccentricities and one other distinct parameter characterizing them are equal.

[9] This can be seen as follows: One can situate one of the vertices with a given angle at the south pole and run the side with given length up the prime meridian.

The two triangles on the left are congruent. The third is similar to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles . The unchanged properties are called invariants .
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'. Note hatch marks are used here to show angle and side equalities.
The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area . (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles.