We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.
Broughton, Sean A.; Haney, Dawn M.; McKeough, Lori T.; and Smith, Brandy M., "Divisible Tilings in the Hyperbolic Plane" (1999). Mathematical Sciences Technical Reports (MSTR). 60.