Editing Congruence (geometry) (section)

===Determining congruence===
[[Image:Congruent triangles.svg|thumb|right|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.]]

Sufficient evidence for congruence between two triangles in [[Euclidean space]] can be shown through the following comparisons:

*'''SAS''' (side-angle-side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
*'''SSS''' (side-side-side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
*'''ASA''' (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

The ASA postulate is attributed to [[Thales of Miletus]]. In most systems of axioms, the three criteria – SAS, SSS and ASA – are established as [[theorem]]s. In the [[School Mathematics Study Group]] system '''SAS''' is taken as one (#15) of 22 postulates.

*'''AAS''' (angle-angle-side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°. ASA and AAS are sometimes combined into a single condition, '''AAcorrS''' – any two angles and a corresponding side.<ref>{{cite book
  | last = Parr
  | first = H. E.
  | title = Revision Course in School mathematics
  | publisher = G Bell and Sons Ltd.
  | series = Mathematics Textbooks Second Edition
  | year = 1970
  | isbn = 0-7135-1717-4}}</ref>
*'''RHS''' (right-angle-hypotenuse-side), also known as  '''HL''' (hypotenuse-leg): If two right-angled triangles have their hypotenuses equal in length, and a pair of other sides are equal in length, then the triangles are congruent.

====Side-side-angle====
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. The opposite side is sometimes longer when the corresponding angles are acute, but it is ''always'' longer when the corresponding angles are right or obtuse. Where the angle is a right angle, also known as the hypotenuse-leg (HL) postulate or the right-angle-hypotenuse-side (RHS) condition, the third side can be calculated using the [[Pythagorean theorem]] thus allowing the SSS postulate to be applied.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the [[ambiguous case]] and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence.

====Angle-angle-angle====
In Euclidean geometry, AAA (angle-angle-angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does not provide information regarding the size of the two triangles and hence proves only [[similarity (geometry)|similarity]] and not congruence in Euclidean space.

However, in [[spherical geometry]] and [[hyperbolic geometry]] (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface.<ref>{{cite book
  | last = Cornel
  | first = Antonio
  | author-link = Antonio Coronel
  | title = Geometry for Secondary Schools
  | publisher = Bookmark Inc.
  | series = Mathematics Textbooks Second Edition
  | year = 2002
  | isbn = 971-569-441-1}}</ref>