.less than
In hyperbolic geometry, triangles have angles that sum to less than 180 degrees, which contrasts with Euclidean geometry where the sum is exactly 180 degrees. This means that while hyperbolic triangles can still have angle measurements in degrees, the total of those angle measures will always be less than 180. Consequently, the concept of "degrees" is applicable, but the properties of the triangles differ significantly from those in Euclidean space.
never In non-Euclidean geometry triangles can have angles that are more or less than 180 degrees.
In an equilateral trianger, each angle is 60 degrees. In all triangles, including equilateral triangles, all 3 angles add up to 180 degrees.(the above assumes Euclidean - flat - space. In Hyperbolic space there are less than 180° in all triangles; in Spherical space more than 180° in all triangles)
180 degrees
A hexagon has 6 triangles inside of it. If one triangle equals 180 degrees and there are 6 triangles. 180 degrees *6 triangles=1080 degrees.
In hyperbolic geometry, triangles have angles that sum to less than 180 degrees, which contrasts with Euclidean geometry where the sum is exactly 180 degrees. This means that while hyperbolic triangles can still have angle measurements in degrees, the total of those angle measures will always be less than 180. Consequently, the concept of "degrees" is applicable, but the properties of the triangles differ significantly from those in Euclidean space.
180 degrees all triangles have 180 degrees.(the above assumes Euclidean - flat - space. In Hyperbolic space there are less than 180° in all triangles; in Spherical space more than 180° in all triangles)
more than
more than
more than
never In non-Euclidean geometry triangles can have angles that are more or less than 180 degrees.
The proof is pretty simple, but hard to see without the pictures. SO here is a link to the proof with some pics. http://www.apronus.com/geometry/triangle.htm The answer depends on your geometry: In Euclidean geometry, the angle sum is 180 degrees, in Hyperbolic geometry it is less than 180 degrees, and in Elliptical geometry it is greater than 180 degrees.
more than
In an equilateral trianger, each angle is 60 degrees. In all triangles, including equilateral triangles, all 3 angles add up to 180 degrees.(the above assumes Euclidean - flat - space. In Hyperbolic space there are less than 180° in all triangles; in Spherical space more than 180° in all triangles)
Only in plane Euclidean geometry and only for triangles
true apex
All triangles have a total of 180 degrees for interior angles in Euclidean plane geometry