We present an approach for handling geometric constraint problems with under-constrained configurations. The approach works by completing the given set of constraints with constraints that can be defined either automatically or drawn from an independently given set of constraints placed on the geometries of the problem. In both cases, the resulting completed set of constraints is not over-constrained. If every well-constrained subproblem in the given under-constrained configuration is solvable, the completed constraint problem is also solvable. Keywords: Constraint solving, geometric constraints, rule-based and graph-based constraint solvers, over- and underconstrained systems.