This paper presents a framework for reasoning about robust geometric algorithms. Robustness is formally defined and a data structure called an approximate polygon is introduced and used to reason about polygons constructed of edges whose positions are uncertain. A robust algorithm for point location in an approximate polygon is presented. The algorithm uses only the signature of the point (not its location) to determine whether the point is inside or outside the polygon. An approximate polygon could, by shifting its edges back and forth within their error bounds, induce a large number of different line arrangements. The cell $C_{a}$ with signature $\alpha$ in one such arrangement will be different than the cell ${C'}{a}$ with signature $\alpha$ in another arrangement. This paper proves that, regardless of their position and shapes, the cells $C{a}$ and ${C'}_{a}$ are always to the same side of the polygons which induce their respective arrangements.