The Triangle Inequality

Without loss of generality we may assume $a\leq b\leq c$.
It is quite obvious that $$a+c\geq b\qquad a+b>c$$
Rotate $\mathcal{BC}$ about $\mathcal{B}$ and $\mathcal{AC}$ about $\mathcal{A}$.

Now as we can see in the above diagram $$a+b>c$$

NOTE:$a+b=c$ would imply that the triangle is degenerate.