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A triangle is a polygon with three sides and three vertices. It is one of the basic shapes in geometry. A triangle with vertices a, B, and C is represented as triangle ABC.

In Euclidean geometry, when any three points are not collinear, a unique triangle and a unique plane (i.e., two-dimensional Euclidean space) are determined. In other words, only one plane contains the triangle, and each triangle is contained in a plane. If the whole geometry is only Euclidean plane, there is only one plane, and all triangles are included in it; however, in high dimensional Euclidean space, this no longer holds. This paper deals with triangles in Euclidean geometry, especially triangles in Euclidean plane, unless otherwise stated.

Triangles are considered 2D planar unless the context indicates otherwise (see non planar triangles below). Therefore, in strict processing, triangles are called 2 simplex (see also polyhedron). In about 300 BC, Euclid introduced the basic facts about triangles in Book 1-4 of the basic principles.

The inner angle of a triangle is always 180 degrees (the same color means they are equal).

In Euclidean space, the sum of internal angles of triangles is always 180 degrees. This fact is equivalent to Euclid's parallel hypothesis. Given a measurement of two angles, this determines the measurement of the third angle of any triangle. The outer angle of a triangle is the angle that is linearly opposite (and therefore complementary) to the inner angle. The measure of the outer angle of a triangle is equal to the sum of two non adjacent inner angles. This is the outer angle theorem. The sum of the three outer angles (one at each vertex) of any triangle is 360 degrees.