Elementary Geometry for College Students (6th Edition)

Using proof by contradiction Suppose $a^{2}+b^{2}= c^{2}$ Then by The pythagorean theorem the triangle is a right triangle this contradicts the given that triangle ABC is not a right triangle. Therefore $a^{2}+ b^{2} \ne c^{2}$.
We know from given theorems that if $a^2 + b^2 < c^2$, then the triangle is obtuse and that if $a^2 + b^2 > c^2$, then the triangle is acute. The only other triangle is a right triangle, and the only other formula is $a^2 + b^2 = c^2$. Thus, a triangle is right if and only if $a^2 + b^2 = c^2$. Thus, if $a^2+b^2 \ne c^2$, then the triangle is not a right triangle.