With this extension, you can apply the Pythagorean Theorem to compare the areas of, say two medium pizza's with one large pizza. Place the diamaters of the medium pizza along the two sides around the angle, and the diameter of the large pizza along the hypotenuse. If the resulting angle is 90 - the area's are equal. If less than 90, the two medium are larger. Here is a second direction.
Some people think that that was the Greek use of the theorem - to constructively add squares.
With that, you can take any polygon, divide it into triangles, change them into squares and add them. The goal is to prove that ANY polygon can be constructively changed into a single square of the same area. This is called 'dissection' and 'equidecomposibility' in the geometry texts. Three of them might be of interest. There is a proof of the Theorem of Pythagoras that comes from an American president.