1. **Problem statement:** Given that $m\overarc{CT} = 110$ and $m\overarc{BT} = 50$, find $m\angle A$ where $AT$ is a tangent to the circle. 2. **Formula and rule:** The angle formed by a tangent and a chord through the point of tangency is half the measure of the intercepted arc. That is, $$m\angle A = \frac{1}{2} m\overarc{BC}$$ where $\overarc{BC}$ is the arc intercepted by the tangent and chord. 3. **Identify the intercepted arc:** The intercepted arc $\overarc{BC}$ is the sum of arcs $\overarc{BT}$ and $\overarc{CT}$: $$m\overarc{BC} = m\overarc{BT} + m\overarc{CT} = 50 + 110 = 160$$ 4. **Calculate the angle:** $$m\angle A = \frac{1}{2} \times 160 = 80$$ 5. **Answer:** $$\boxed{80^\circ}$$ This means the angle formed by the tangent $AT$ and chord $BT$ is $80$ degrees.