This is a question from Hartshorne's Geometry: Euclid and Beyond. For this problem you are supposed to use the usual congruence axioms and notions of betweenness as well as the incidence axioms.
Show that addition preserves inequalities: Let $AB$ and $CD$ be two line segments. If $AB<CD$ and $EF$ is any other line segment then $AB+EF<CD+EF$.
What I have done so far is basically said that since $AB < CD$ we have that for some point $x$ between $CD$ it is the case that $AB$ is congruent to $CX$. From here can I use the fact that $AB<CD+EF$ in general and implement another line to somehow get the relation?
Thanks
