1. **Problem:** Given $m \angle AFD = 120^\circ$ and $m \angle BE = 50^\circ$, find $m \angle ACD$. 2. **Understanding the problem:** Points $A$, $B$, $C$, $D$, and $E$ lie on or outside a circle with secants intersecting at $C$. We want to find the measure of angle $ACD$ formed by these secants. 3. **Formula used:** The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs: $$m \angle = \frac{1}{2} |m \text{arc}_1 - m \text{arc}_2|$$ 4. **Identify arcs:** Here, $m \angle AFD = 120^\circ$ and $m \angle BE = 50^\circ$ correspond to arcs intercepted by the secants. 5. **Calculate $m \angle ACD$:** $$m \angle ACD = \frac{1}{2} |120^\circ - 50^\circ| = \frac{1}{2} \times 70^\circ = 35^\circ$$ 6. **Answer:** $$\boxed{35^\circ}$$