In this paper, we study two-dimensional Riemann boundary value problems of Euler system for the isentropic and irrotational Chaplygin gas with initial data being two constant states given in two sectors respectively, where one sector is a quadrant and the other one has an acute vertex angle. We prove that the Riemann boundary value problem admits a global self-similar solution, if either the initial states are close, or the smaller sector is also near a quadrant. Our result can be applied to solving the problem of shock reflection by a ramp.