Here is exercise 10 from section 4-4 out of M. do Carmos's Differential Geometry of Curves and Surfaces:
Show that the geodesic curvature of an oriented curve $C\subset S$ at a point $p\in C$ is equal to the curvature of the plane curve obtained by projecting $C$ onto the tangent plane $T_p(S)$ along the normal to the surface at $p$.
As a tip for this exercise, the autor writes:
Apply the relation $k_g^2 + k_n^2 = k^2$ and the Meusnier theorem to the projecting cylinder.
First, I don't see how the Meusnier theorem applies here, since it deals with curves that are on the same surface. Besides, what cylinder is he talking about?