Sum of angles of a triangle


In a Euclidean space, the sum of angles of a triangle equals the straight angle.
A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.
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 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.

Cases

Euclidean geometry

In Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. In the presence of the other axioms of Euclidean geometry, the following statements are equivalent:
The sum of the angles of a hyperbolic triangle is less than 180°. The relation between angular defect and the triangle's area was first proven by Johann Heinrich Lambert.
One can easily see how hyperbolic geometry breaks Playfair's axiom, Proclus' axiom, the equidistance postulate, and Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails.
The sum of the angles can be arbitrarily small. For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal to zero.

Spherical geometry

For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°. Specifically, the sum of the angles is
where f is the fraction of the sphere's area which is enclosed by the triangle.
Note that spherical geometry does not satisfy several of Euclid's axioms

Exterior angles

Angles between adjacent sides of a triangle are referred to as interior angles in Euclidean and other geometries. Exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the sum of all three exterior angles, that equals to 360° in the Euclidean case, is less than 360° in the spherical case, and is greater than 360° in the hyperbolic case.

In differential geometry

In the differential geometry of surfaces, the question of a triangle's angular defect is understood as a special case of the Gauss-Bonnet theorem where the curvature of a closed curve is not a function, but a measure with the support in exactly three points – vertices of a triangle.