The sum of the angles of a triangle is exactly 180° in Euclidean geometry, greater than 180° in spherical (or elliptical/Riemannian) geometry, and less than 180° in hyperbolic geometry.
Conditions: The figure is a triangle.; The geometry is specified as Euclidean, spherical, or hyperbolic.
The sum of the angles of a triangle is exactly 180° in Euclidean geometry, greater than 180° in spherical (or elliptical/Riemannian) geometry, and less than 180° in hyperbolic geometry.
Conditions: The figure is a triangle.; The geometry is specified as Euclidean, spherical, or hyperbolic.
In spherical geometry, there are no straight lines through a point not on a given line that are parallel to that line. This is because the surface is positively curved, causing all straight lines (geodesics) to eventually intersect.
Conditions: The geometry is spherical geometry.; There is a given straight line.; There is a point not on that line.
In spherical geometry, there are no straight lines through a point not on a given line that are parallel to that line. This is because the surface is positively curved, causing all straight lines (geodesics) to eventually intersect.
Conditions: The geometry is spherical geometry.; There is a given straight line.; There is a point not on that line.