In hyperbolic geometry, an ideal triangle is a set of three lines which connect three distinct points on the boundary of the model of hyperbolic geometry.

Below is an example of an ideal triangle in the Beltrami-Klein model:

Below is an example of an ideal triangle in the Poincaré disc model:

Below are some examples of ideal triangles in the upper half plane model:

Strictly speaking, none of these figures are triangles in hyperbolic geometry; however, ideal triangles are useful for proving that, given $r \in \mathbb{R}$ with $0<r<\pi$ , there is a triangle in hyperbolic geometry whose angle sum in radians is equal to $r$ .