We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NP-completeness of deciding whether there is such a perfect matching. For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge. We also construct polygons with maximal compatible matchings with n/7 edges, demonstrating the tightness of this bound. Tight bounds on the size of a minimal maximal compatible matching are also obtained for the families of d-regular geometric graphs for each d in {0,1,2}. Finally we consider a related problem. We prove that it is NP-complete to decide whether a noncrossing geometric graph G admits a set of compatible noncrossing edges such that G together with these edges has minimum degree five.