A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem x ′′ = f(t, x, x′ ), x(0) = 0, x ′ (1) = ∫ 1 0 x ′ (s) dg(s), with g increasing on [0, 1] such that ∫ 1 0 dg = 1, is a couple of functions σ, τ ∈ C 1 ([0, 1]) such that σ(t) ≤ τ (t) for all t ∈ [0, 1], σ ′ (t) ≥ f(t, x, σ(t)), σ(1) ≤ ∫ 1 0 σ(s) dg(s), τ ′ (t) ≤ f(t, x, τ (t)), τ (1) ≥ ∫ 1 0 τ (s) dg(s), in the stripe ∫ t 0 σ(s) ds ≤ x ≤ ∫ t 0 τ (s) ds and t ∈ [0, 1]. It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.