For a harmonic map (Formula presented.) transforming the contour of an angle of the boundary ∂Z into a rectilinear segment of the boundary ∂W, the behavior near the vertex of the specified angle is investigated. The behavior of the inverse map (Formula presented.) near the preimage of the vertex is investigated as well. In particular, we prove that if ϕ is the value of the angle at which a ray λ ϕ is issued from the vertex, and θ is the value of the angle at which its image F(λ ϕ ) leaves the vertex's, then the dependence of θ on ϕ is described by a discontinuous function. Thus, such a behavior of the harmonic map qualitatively differs from the behavior of the corresponding conformal map: for the latter one, the dependence θ(ϕ) is described by a linear function. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.