In this paper, the authors study the boundary controllability of a class of one-dimensional degenerate equations. The novelty of this work is that the control acts through the part of the boundary where degeneration occurs. First, the authors consider a class of degenerate hyperbolic equations. Then, they prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.