Editing Triangle (section)

=== Angles ===
[[File:Triangle sommeangles.svg|thumb|The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).]]
The [[Sum of angles of a triangle|sum of the measures of the interior angles of a triangle]] in [[Euclidean space]] is always 180 degrees.{{sfn|Heath|1926|loc=Proposition 32}} This fact is equivalent to Euclid's [[parallel postulate]]. This allows the determination of the measure of the third angle of any triangle, given the measure of two angles.{{sfn|Gonick|2024|pages=107–109}} An ''[[exterior angle]]'' of a triangle is an angle that is a linear pair (and hence [[supplementary angle|supplementary]]) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the [[exterior angle theorem]].{{sfn|Ramsay|Richtmyer|1995|p=[https://books.google.com/books?id=4CDpBwAAQBAJ&pg=PA38 38]}} The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has.{{sfn|Gonick|2024|pages=224–225}}

Another relation between the internal angles and triangles creates a new concept of [[trigonometric function]]s. The primary trigonometric functions are [[sine and cosine]], as well as the other functions. They can be defined as the [[Sine and cosine#Right-angled triangle definition|ratio between any two sides of a right triangle]].{{sfn|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}} In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the [[law of sines]] and the [[law of cosines]].{{sfn|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}}

Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A [[Degeneracy (mathematics)#Triangle|degenerate triangle]], whose vertices are [[collinearity|collinear]], has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.<ref>{{cite journal
 | last1 = Richmond | first1 = Bettina | author1-link = Bettina Richmond
 | last2 = Richmond | first2 = Thomas
 | doi = 10.1080/00029890.1997.11990706
 | issue = 8
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2975234
 | mr = 1476755
 | pages = 713–719
 | title = Metric spaces in which all triangles are degenerate
 | volume = 104
 | year = 1997}}</ref><ref>{{cite thesis|type=PhD thesis|title=Making sense of definitions in geometry: Metric-combinatorial approaches to classifying triangles and quadrilaterals|last=Alonso|first=Orlando Braulio|publisher=Teachers College, Columbia University|year=2009|id={{ProQuest|304870039}}|page=57}}</ref>) The conditions for three angles <math> \alpha </math>, <math> \beta </math>, and <math> \gamma </math>, each of them between 0° and 180°, to be the angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles <math> \alpha </math>, <math> \beta </math>, and <math> \gamma </math> exists [[if and only if]]<ref>{{multiref
|{{harvnb|Verdiyan|Salas|2007}}
|{{harvnb|Longuet-Higgins|2003}}
}}</ref>
<math display="block"> \cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma) = 1.</math>