Why Can we Limit the Cases of Some Triangles?

G4n (n ≥2)

In G4n there are vertices with 2 α angles and 2 β angles. At a four way intersection the angles must be equal and are then 90 degree angles. So, α = β = 90.

I8n

In the I8n tilings there are vertices with 4 β and 2 γ angles. They will only be divisions if β = γ. This only occurs for n = 3 with β = γ = 60°. That is the I24 case.