The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which are not right.



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why most of the angles are right
I'd be very surprised if one can do better than just reading part of the proof of the classification, which is a beautiful combinatorial argument already, in my opinion.


For example, in the proof of the classification of irreducible root systems given in Fulton and Harris (of Theorem 21.11), part (iv) says that you can collapse a long string of nodes in the diagram if the endpoints are connected to any other nodes. This will inductively force diagrams with more than one set of multiple lines (i.e. non-right angles) to be quite small.


