Common Definition 1: If $A$, $B$, and $C$ are sides of a triangle, then $|A+B| \leq |A| + |B|$. Intuition: We are able to provide an upper bound for the third side of the triangle which is really only met in the degenerate case (when the area of the triangle is 0). Still don't really get it? (so that's why I'm asking)
Common Definition 2: If $x, y, z \in \mathbb{R}$, then $|x-y| \leq |x-z| + |z-y|$
Intuition: For any triangle, the length of any side cannot exceed the sum of the lengths of the other two sides.