Ha!
If you have a measuring device capable of measuring distance and angles to a significant degree of "real world" accuracy will the Pythagorean Theorem apply to the real world, describe reality to the same degree of accuracy as you are capable of?
There is no such thing as a perfect right angle triangle in nature. The Pythagorean Theorem describes the real world in ideal terms. Spheres approximate celestial bodies such as the Moon, Mars, etc. but these are not perfect spheres, so all Euclidean geometry is a schematic approximation of the real world and consequently the ratios obtained through it are approximations as well.
The Pythagorean Theorem is true only in triangles that have a perfect 90 degree angle. In all other cases it is an appxorimation. But where would you draw the line? At 89, 88, 87, 82.567 degrees for an "acceptable" approximation? The Theorem is theoretically true only for theoretically perfect triangles.
Look, a perfect parabolic mirror, in theory, will focus all the light coming from a distance point source to a perfect point. But, there is no such thing as a perfect parabolic mirror and the nature of light makes it impossible to concentrate all the energy into a point, even if the mathematics say it can. Nature trumps math every time.
Using these measurements and mathematics can you prove it in the real world? Why?
Prove what? That the hypotenuse value squared is exactly the sum of the squares of the sides? No.
Yes. To whatever degree of accuracy you use.
No.
Call Scientific American, the Pythagorean Theorem has been revoked.
Just when I thought you couldn't dig any deeper...