jmc

Angle-chasing using the small triangles: Use the line below and to the left of the $110^{\circ }$ angle to find that the rightmost angle in the small lower-left triangle is $180-110=70^{\circ }$. Then use the small lower-left triangle to find that the remaining angle in that triangle is $180-70-40=70^{\circ }$. Use congruent vertical angles to find that the lower angle in the smallest triangle containing $A$ is also $70^{\circ }$. Next, use line segment $AB$ to find that the other angle in the smallest triangle containing $A$ is $180-100=80^{\circ }$. The small triangle containing $A$ has a $70^{\circ }$ angle and an $80^{\circ }$ angle. The remaining angle must be $180-70-80=\boxed{30}$