The dihedral angle between two of the root hyperplanes in a root system can be $90$ , $60$ , $45$ or $30$ degrees and nothing else.
Dihedral angles
Dihedral angles
Statement of result
Explanation
By the angle between two hyperplanes, I mean the following:

take the unit vectors $v$ and $w$ orthogonal to these hyperplanes,

take the smallest angle $\varphi $ such that $\mathrm{cos}\varphi =v\cdot w$ .
In a root system, the vectors orthogonal to the root hyperplanes are the (simple) roots, so we are really interested in $\alpha \cdot \beta $ where $\alpha $ and $\beta $ are roots. Recall that the dot product is the Killing form.
Proof
Since we're in a root system, if $\alpha $ and $\beta $ are roots then ${P}_{\alpha}(\beta )=\frac{1}{2}{n}_{\beta \alpha}\alpha $ for some ${n}_{\beta \alpha}\in \mathbf{Z}$ , where ${P}_{\alpha}(\beta )$ is the orthogonal projection of $\beta $ onto the line through $\alpha $ .
In the $SU(3)$ root system shown below, the projection of $\beta $ to the line through $\alpha $ gives $\alpha /2$ .
By Euclidean geometry, we have $${P}_{\alpha}(\beta )=\frac{\alpha \cdot \beta}{\alpha \cdot \alpha}\alpha .$$
This means that $${n}_{\beta \alpha}=\frac{2\alpha \cdot \beta}{\alpha \cdot \alpha}.$$ We also have $${n}_{\alpha \beta}=\frac{2\alpha \cdot \beta}{\beta \cdot \beta},$$ so $${n}_{\beta \alpha}{n}_{\alpha \beta}=\frac{4{(\alpha \cdot \beta )}^{2}}{(\alpha \cdot \alpha )(\beta \cdot \beta )}=4{\mathrm{cos}}^{2}(\varphi ),$$
where $\varphi $ is the angle between the roots.
We have $$ because the roots are not collinear, so $${n}_{\alpha \beta}{n}_{\beta \alpha}=4{\mathrm{cos}}^{2}\varphi \in \{0,1,2,3\}$$ because it's an integer between 0 and 4.
These possibilities give $\varphi =\{\begin{array}{cc}90\hfill & \\ 60\hfill & \\ 45\hfill & \\ 30\hfill & \end{array}$ degrees.