Back Виçкĕтеслĕхĕн кĕтесĕсен суммин теореми CV Vinkelsum Danish Winkelsumme German Suma de los ángulos de un triángulo Spanish Somme des angles d'un triangle French סכום הזוויות במשולש HE Եռանկյան անկյունների գումար Armenian Hornasumma þríhyrnings Icelandic Trigésima segunda proposição de Euclides Portuguese Теорема о сумме углов треугольника Russian
In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.
The sum can be computed directly using the definition of angle based on the dot product and trigonometric identities, or more quickly by reducing to the two-dimensional case and using Euler's identity.
It was unknown for a long time whether other geometries exist, for which 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.
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