1. **State the problem:** We are given a polygon with points A, B, C, D, E, F, G and certain lengths and angle measures: $EG=3$, $EB=9$, $AF=8$, $m\angle EBG=19$, $m\angle EGF=29$, and $m\angle CAE=50$. We need to find $m\angle CAF$. 2. **Analyze the given information:** The angles $\angle EBG$ and $\angle EGF$ are parts of triangle $EBG$. The angle $\angle CAE$ is given, and we want to find $\angle CAF$, which shares vertex $A$ and points $C$ and $F$. 3. **Use angle addition:** Since $\angle CAF$ and $\angle CAE$ share vertex $A$ and side $AC$, and points $E$ and $F$ lie on segments connected to $A$, we can express $m\angle CAF$ as the sum or difference of known angles. 4. **Calculate $m\angle EGF$ and $m\angle EBG$ sum:** $$m\angle EBG + m\angle EGF = 19 + 29 = 48$$ 5. **Use triangle $EBG$ angle sum:** The sum of angles in triangle $EBG$ is $180$ degrees. $$m\angle BEG = 180 - (19 + 29) = 180 - 48 = 132$$ 6. **Relate $m\angle BEG$ to $m\angle CAE$ and $m\angle CAF$:** Given the polygon and the red marks indicating equal segments, $m\angle CAF$ can be found by subtracting $m\angle CAE$ from $m\angle BEG$. 7. **Calculate $m\angle CAF$:** $$m\angle CAF = m\angle BEG - m\angle CAE = 132 - 50 = 82$$ **Final answer:** $$m\angle CAF = 82$$