True.
Let's call the sides of the two right triangles as follows:
Triangle 1: legs a,b, hypothenuse c.
Triangle 2: legs x,y, hypothenuse z.
Suppose we know that a=x and c=z. By Pythagorean theorem, we have for both triangles
[tex] \text{hypotenuse}^2 = \text{leg}_1^2 + \text{leg}_2^2 [/tex]
So, in triangle 1 we have
[tex] c^2 = a^2+b^2 [/tex]
while in triangle 2 we have
[tex] z^2 = x^2+y^2 [/tex]
But since a=x and c=z, the second equation becomes
[tex] c^2 = a^2+y^2 [/tex]
Solve the equations for y^2 and b^2: we have in both cases
[tex] y ^2 = b^2 = c^2-a^2 [/tex]
This, in theory, would mean[tex] |b|=|y| [/tex], but they are length of a triangle, and thus are both positive, which concludes the proof.
