section_path: Properties/Angles
section_name: Angles
paragraph_text: Angles
  
  The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. 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. An exterior angle of a triangle is an angle that is a linear pair (and hence 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. 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.
  
  Another relation between the internal angles and triangles creates a new concept of trigonometric functions. The primary trigonometric functions are sine and cosine, as well as the other functions. They can be defined as the ratio between any two sides of a right triangle. 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.
  
  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 degenerate triangle, whose vertices are collinear, has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.) The conditions for three angles α {\displaystyle \alpha }, β {\displaystyle \beta }, and γ {\displaystyle \gamma }, 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 α {\displaystyle \alpha }, β {\displaystyle \beta }, and γ {\displaystyle \gamma } exists if and only if cos 2 α + cos 2 β + cos 2 γ + 2 cos (α) cos (β) cos (γ) = 1. {\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha)\cos(\beta)\cos(\gamma)=1.}