If one triangle has angles that summed to more 180', there are three possible cases:
The angles of every triangle add up to 180' exactly (Euclidian Geometry)
The angles of every triangle add up to less than 180'
The angles of every triangle add up to more than 180' (a case that Legendre thought he had excluded; it later turned out that he had made other unstated assumptions to do so)
Hypothesis of the obstuse angle: both C and D are greater than a right angle
Hypothesis of the acute angle: both C and D are less than a right angle
The universe as a whole has positive curvature and its topology is that of a sphere
There are many alternatives to Euclid's geometry, and that the nature of physical space is a question of observation, not thought alone.
The angles of every triangle add up to 180' exactly (Euclidian Geometry)
The angles of every triangle add up to less than 180'
The angles of every triangle add up to more than 180' (a case that Legendre thought he had excluded; it later turned out that he had made other unstated assumptions to do so)
Hypothesis of the obstuse angle: both C and D are greater than a right angle
Hypothesis of the acute angle: both C and D are less than a right angle
The universe as a whole has positive curvature and its topology is that of a sphere
There are many alternatives to Euclid's geometry, and that the nature of physical space is a question of observation, not thought alone.