We are told that $(x,y)\in R$ iff $|x+y|=|x|+|y|$ and that $x,y \in \mathbb{R}$.

I've created the two statements:
$$(x,y) \in R \Rightarrow|x+y| = |x|+|y|$$
$$(y,z) \in R \Rightarrow|y+z| = |y|+|z|$$
But im not sure on how to proceed. I know I have to test for $(x,z) \in R \Rightarrow|x+z| = |x|+|z|$ i tried subbing the two equations into $(x,z)$, but I had no luck with that. Any thoughts?