In a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

, the sum of angles of a triangle equals the straight angle (180 degrees,

radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

, two

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...

s, or a half- turn). A

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

has three angles, one at each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

, 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 (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 In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the def ...

'' and serves as an important distinction for geometric systems.

Cases

Euclidean geometry

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, the triangle postulate states that the sum of the angles of a triangle is two

s. This postulate is equivalent to the

parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...

. In the presence of the other axioms of Euclidean geometry, the following statements are equivalent: *Triangle postulate: The sum of the angles of a triangle is two right angles. *

Playfair's axiom In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): ''In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the ...

: Given a straight line and a point not on the line, exactly one straight line may be drawn through the point parallel to the given line. * Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also. *Equidistance postulate: Parallel lines are everywhere equidistant (i.e. the

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

from each point on one line to the other line is always the same.) *Triangle area property: The

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

of a triangle can be as large as we please. *Three points property: Three points either lie on a line or lie on a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

. *

Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Hyperbolic geometry

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 Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...

. One can easily see how

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

, so the three points property also fails. The sum of the angles can be arbitrarily small (but positive). For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal to zero.

Spherical geometry

For a

spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...

, the sum of the angles is greater than 180° and can be up to 540°. Specifically, the sum of the angles is :180° × (1 + 4''f'' ), 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 (including the

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°From the definition of an exterior angle, its sums up to the straight angle with the interior angles. So, the sum of three exterior angles added to the sum of three interior angles always gives three straight angles. in the Euclidean case (as for any

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

), 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 In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...

, 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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

is not a function, but a measure with the

support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

in exactly three points – vertices of a triangle. {{expand section, date=November 2013

References

Geometry Triangle geometry