Modification of the triangle inequality

Question

We know from the triangle inequality that $X+Y \geq Z$,

My question is under what conditions of $a,b,c$ (acute, obtuse or right angle) that $Z >X $ and $Z \geq Y $

Answer 1

The sine rule tells you that

$$\frac{Z}{\sin c} = \frac{X}{\sin a}$$

So $Z > X$ iff $\sin c > \sin a$. Since $0 < a$, $0 < c$ and $a + c < \pi$, $\sin c > \sin a$ iff $c > a$.

Similiarly $Z \geq Y$ iff $c \geq b$