Converse of the Pythagorean Theorem
The converse of the Pythagorean theorem states that if $a^2 + b^2 = c^2$ for three sides of a triangle, then the triangle is a right triangle.
Formula
\text{If } a^2 + b^2 = c^2 \text{, then the triangle is a right triangle}
Definition
The converse of the Pythagorean theorem says that if three sides of a triangle satisfy $a^2 + b^2 = c^2$ (with $c$ largest), the triangle must have a right angle opposite $c$; if $a^2 + b^2 > c^2$, the triangle is acute, and if $a^2 + b^2 < c^2$, it is obtuse. It is provable by the SSS congruence criterion: constructing a right triangle with legs $a$, $b$ gives hypotenuse $\sqrt{a^2+b^2} = c$ by construction, so SSS gives congruence to the original triangle. The result fails in non-Euclidean geometry.
Example
Sides $5$, $12$, $13$: $5^2 + 12^2 = 169 = 13^2$, confirming a right triangle, the basis of the carpenter's $3$-$4$-$5$ rule for checking right angles. Sides $7$, $9$, $12$: $7^2 + 9^2 = 130$ versus $12^2 = 144$, and since $130 < 144$, the triangle is obtuse. In spherical geometry, a triangle with vertices at $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ on the unit sphere has three right angles and equal sides of $\pi/2$, yet $(\pi/2)^2 + (\pi/2)^2 \neq (\pi/2)^2$, so the Euclidean converse fails there.
Key Insight
The law of cosines, $c^2 = a^2 + b^2 - 2ab\cos(C)$, connects all three cases: when $C = 90^\circ$, $\cos(C) = 0$ and we recover the Pythagorean theorem, and the sign of $a^2 + b^2 - c^2$ tells us whether $C$ is $90^\circ$, acute, or obtuse. The failure of the converse in non-Euclidean geometry is precisely what makes the Pythagorean theorem equivalent to the parallel postulate; any geometry satisfying $a^2 + b^2 = c^2$ for all right triangles must be Euclidean flat geometry.