# Is the sum of coroots the coroot of the sum?

While studying the book Introduction to Lie Algebras and Representation Theory, by Humphreys, I've came across a problem that seems simple, but I just cannot figure out:

If $$\alpha, \beta \in \Phi$$, where $$\Phi$$ is the root system, then $$(\alpha + \beta)^\vee = \alpha ^\vee + \beta^\vee$$?

Studying the algebras I already know it feels true, but I wasn't able to show a proper proof.

Any help would be appreciated.

If this were true, a root system would be isomorphic to its dual, which need not be the case. Look for a counterexample in type $$B$$.