jmc

A diagram will probably help.



Since we are considering the incenter, $AY=AZ$, and likewise around the triangle. Therefore the three outer triangles are isosceles.



This lets us determine two of the angles at $Z$:



Therefore $$\angle YZX=180^{\circ }-50^{\circ }-70^{\circ }=\boxed{60^{\circ }}.$$