We prove that any PSLG has a conforming quadrilateral mesh with O(n) elements, all angles ≤ 120◦ and all new angles ≥ 60◦ (the complexity and the angle bounds are both sharp). Moreover, all but O(n) of the angles may be taken in a smaller interval, say [89◦, 91◦]. Date: January 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: