Final answer:
SAS is approximatively true for 'sufficiently small triangles' on curved surfaces such as spheres, where 'small' means the curvature between vertices is negligible, allowing local spherical geometry to mimic Euclidean geometry. For the Earth, small sections are often approximated as flat. The volume of a sphere is given by 4³/3, not 4².
Step-by-step explanation:
If Side-Angle-Side (SAS) is not true for all triangles on a sphere, it might still hold true for sufficiently small triangles. One way to think about this is to consider what we mean by 'small triangles' in the context of a curved surface like a sphere. As triangles get smaller, their vertices get closer together and the curvature of the sphere between these vertices becomes negligible. Thus, a small triangle could be defined as a triangle that is small enough such that the curvature of the sphere does not significantly affect the triangle and SAS can be applied as it would in Euclidean geometry.
When we talk about the Earth and large surfaces, we can approximate small sections as flat. We do this all the time with our local surroundings. Empirical observations and consistent truths give us a basis to apply logical rules, such as those governing triangles on a spherical surface. When the curvature is negligible, the local geometry is essentially Euclidean. Therefore, SAS would be applicable to these 'small' or 'local' triangles.
The formula for the volume of a sphere is 4³/3, while the expression 4² represents the surface area of a sphere of radius r.