Editing Triangle (section)

=== Similarity and congruence ===
[[File:Angle-angle-side_triangle_congruence.svg|thumb|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_mark#Congruency_notation|hatch marks]] are used here to show angle and side equalities.]]
Two triangles are said to be ''[[similarity (geometry)|similar]]'', if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.{{sfn|Gonick|2024|pages=157–167}}

Some basic [[theorem]]s about similar triangles are:
* [[If and only if]] one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar.{{sfn|Gonick|2024|page=167}}
* If and only if one pair of corresponding sides of two triangles are in the same proportion as another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar.{{sfn|Gonick|2024|page=171}} (The ''included angle'' for any two sides of a polygon is the internal angle between those two sides.)
* If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.{{efn|1=Again, in all cases "mirror images" are also similar.}}

Two triangles that are [[Congruence (geometry)|congruent]] have exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. This is a total of six equalities, but three are often sufficient to prove congruence.{{sfn|Gonick|2024|page=64}}

Some individually [[necessary and sufficient condition]]s for a pair of triangles to be congruent are:{{sfn|Gonick|2024|pages=65,72–73,111}}
* SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
* ASA: Two interior angles and the side between them in a triangle have the same measure and length, respectively, as those in the other triangle. (This is the basis of [[Triangulation (surveying)|surveying by triangulation]].)
* SSS: Each side of a triangle has the same length as the corresponding side of the other triangle.
* AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as ''AAcorrS'' and then includes ASA above.)