For someone in high school, I think it's good to prove that the sum of the interior angles of a triangle is $\pi$ if they don't know why. Personally, I was never shown why this fact is true, and I feel that it's generally a bad idea to not know why something in math is true, especially when the answer is pretty. My favorite proof is to think about how the normal vector changes as you walk around the triangle -- it's nice because it generalizes to other shapes (which may not even be polygons).