Have you already read the article on Wiki on non-Euclidean geometry?GeometryIsHARD said:So it has come to my attention that 'different geometries' have different rules for the interior angles of a triangle... What are these different rules for Euclidean, elliptic and hyperbolic geometry? What I'm really wondering though is what knowing these rules about the interior angles tells us about the extiorior angles. does the interior angle + exterior angle have to equal 180 degree's or something? Thanks for anyone with some insight!
A triangle in Euclidean geometry is a polygon with three sides and three angles. It is formed by three line segments that intersect at three points called vertices.
In non-Euclidean geometries, triangles are classified based on the sum of their interior angles. In hyperbolic geometry, the sum of the interior angles is less than 180 degrees, while in elliptic geometry, the sum is greater than 180 degrees.
Yes, in non-Euclidean geometries, triangles can have more or less than three sides. In hyperbolic geometry, triangles can have more than three sides, while in elliptic geometry, triangles can have less than three sides.
The Pythagorean theorem and other trigonometric formulas may differ in non-Euclidean geometries due to the different definitions of distance and angles. In hyperbolic geometry, the Pythagorean theorem is modified to account for the curvature of space, while in elliptic geometry, the Pythagorean theorem does not apply at all.
Yes, triangles with the same side lengths and angles can exist in different geometries, although their properties may be different. For example, in hyperbolic geometry, these triangles will have larger areas and their interior angles will add up to less than 180 degrees, while in elliptic geometry, these triangles will have smaller areas and their interior angles will add up to more than 180 degrees.