in pythagorean therorem's proof #3, how do we know a square is created with the four right trianges?

Question

 I need to know why we know/how we know that a square is created when we take four right triangles and rotate them around respectfully. it appears to be a square, and I understand we use the C^2 formula....however....we can't assume it is a square, so how do I know it is a square.

Answer 1

 

You know the large figure has 90^o corners because each corner is composed of the two non-right angles from two of the 4 right triangles: look at one of the right triangles, call those angles θ and β:   θ +  β + 90^o = 180^o    so θ + β = 90^o   

 

Of course, the sides are all equal to c, the hypotenuse of each of the right triangles.   Thus, the outer figure is a square.

 

Answer 2

Assuming that you are rotating the four right triangles so that the right angles are the corners of the "square.'

The square is formed by the hypotenuse of each triangle. They are equal. This sides of the "square" are equal making it a square!