  The plane waveguides with corners considered here are infinite V-shaped
strips with constant thickness. They are parametrized by their sole opening
angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when
this angle tends to 0. We provide multi-scale asymptotics for eigenpairs
associated with the lowest eigenvalues. For this, we investigate the eigenpairs
of a one-dimensional model which can be viewed as their Born-Oppenheimer
approximation. We also investigate the Dirichlet Laplacian on triangles with
sharp angles. The eigenvalue asymptotics involve powers of the cube root of the
angle, while the eigenvector asymptotics include simultaneously two scales in
the triangular part, and one scale in the straight part of the guides.
