Is there a net of a sphere?

The answer is NO, as far as I know. vsauce told me that Carl Friedrich Gauss was the first to prove that it is impossible to make a flat map of the earth that does not distort shape or size. If nobody could prove it until Gauss came around, I'd assume it must be a pretty complicated proof. But I recently came up with something:

Suppose that there exists a way to cut a sphere such that it can be laid flat on a plane, and that all straight lines remain straight lines, and all areas, lengths, and angles remain the same. We are also going to

assume that the flattened map can be folded back up into a sphere also. Then, on the flattened sphere, draw a triangle. What do it's angles add up to? 180 degrees. Now fold it back into the sphere. All the line

segments remain line segments, but the angles cannot remain the same measure since they now add up to strictly more than 180 degrees.

In fact, if it's true that all the angles in an equilateral triangle are congruent on a sphere, then there's a way of proving this without using the angles: in the net, draw six equilateral triangles with a common

vertex. If you fold it into a sphere, the result cannot be six equilateral triangles with a common vertex, because the angles must strictly be greater than 60 degrees, and six of them add up to more than 360

degrees and it cannot fit around a point. Therefore, these figures can no longer be equilateral triangles.

Any bending and folding of paper will keep straight lines straight, angles the same, and distances and areas, no?

Is there something wrong with this? Or is it simply that the geometry of a sphere is strictly different from the geometry of a plane?