In which theorem are the angles of a spherical triangle equivalent to those of a plane triangle with increased sides?

The theorem that relates the angles of a spherical triangle to those of a plane triangle with increased sides is known as Legendre's theorem. This theorem is crucial in spherical geometry, where the measures of triangles differ from those in plane geometry due to the curvature of the sphere.

In spherical triangles, the sum of the angles exceeds 180 degrees, unlike in plane triangles where it is always exactly 180 degrees. Legendre's theorem helps to express this relationship by establishing that as the sides of a spherical triangle increase, their corresponding angles behave similarly to those of a plane triangle, but adjusted to account for the geometry of a sphere.

Understanding this theorem is essential for geodetic engineers, as they often work with spherical models of the Earth when performing triangulation and other surveying calculations across large distances. The implications of this theorem impact how angles and distances are calculated on the Earth's surface, making it a fundamental aspect of spherical geometry in geodetic practice.