Quadrilateral meshes for PSLGs

We prove that every planar straight line graph with $n$ vertices has a conforming quadrilateral mesh with $O(n^2)$ elements, all angles $\leq 120^\circ$andallnewangles$\geq 60^\circ$. Both the complexity and the angle bounds are sharp. Moreover, all but $O(n)$ of the angles may be taken in a smaller interval, say $[{89}^{\circ },{91}^{\circ }]$.