Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> = 2 (\alpha,\beta)/(\beta,\beta)$ is an integer, we can conclude that $4 \cos^2(\theta) = <\alpha,\beta><\beta,\alpha> \in \{0,1,2,3\}$. The value 4 cannot occur since $\alpha \neq \pm \beta$.
In Murphey's it is for example written that for $<\alpha,\beta>=<\beta,\alpha> = 0$ it follows that $\theta = \pi / 2$. I am wondering why it cannot be $3\pi /2$ ? Does it follow from the assumptions that $\theta \in (0, \pi)$ ?
We also have the formula $<\alpha,\beta>= 2 \cos \theta ||\alpha||/||\beta||$.
The angle between two vectors is never larger than $\pi$. There are two directions to measure the angle and we pick the smaller one.