In this paper, we study geometric properties of quotient spaces of proper Lie
groupoids. First, we construct a natural stratification on such spaces using an
extension of the slice theorem for proper Lie groupoids of Weinstein and Zung.
Next, we show the existence of an appropriate metric on the groupoid which
gives the associated Lie algebroid the structure of a singular riemannian
foliation. With this metric, the orbit space inherits a natural length space
structure whose properties are studied. Moreover, we show that the orbit space
of a proper Lie groupoid can be triangulated. Finally, we prove a de Rham
theorem for the complex of basic differential forms on a proper Lie groupoid.Comment: 35 pages, minor changes, added reference and remark 3.1