হিঞ্জের উপপাদ্য অথবা (অথবা, SAS অসমতা)

Suppose that we have two triangles, $\triangle ABC$ and $\triangle DEF$. Suppose that we know that $AB=DE$, $AC=DF$, and $m\angle A\gt m\angle D$.

The SAS Inequality Theorem (informally known as the Hinge Theorem) states that $BC\gt EF$.

The intuition for this theorem lies fully in its informal name. If a hinge is opened with a greater angle, then naturally the distance between the two ends is greater, even though the other side lengths are the same.

An incredibly quick proof of this theorem lies in the Law of Cosines.

By the Law of Cosines, $$BC^2=AB^2+AC^2-2AB\cdot AC\cos A$$ $$EF^2=DE^2+DF^2-2DE\cdot DF\cos A$$ Since the cosine function is decreasing on the interval $[0^\circ,180^\circ]$, we know that $\cos A\lt\cos D$.

$AB=DE$ and $AC=DF$, $BC^2\gt EF^2$, so we know that $BC\gt EF$.