Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are $\alpha, \alpha, \beta$ and $\gamma$ in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also $\alpha, \alpha, \beta$ and $\gamma$ in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.