<P> In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides . In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, π radians, two right angles, or a half - turn . </P> <P> It was unknown for a long time whether other geometries exist, where this sum is different . The influence of this problem on mathematics was particularly strong during the 19th century . Ultimately, the answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle . Its difference from 180 ° is a case of angular defect and serves as an important distinction for geometric systems . </P>

The sum of the interior angles of a triangle is