1. **State the problem:** We are given three vectors $\overrightarrow{AB}$, $\overrightarrow{CD}$, and $\overrightarrow{CE}$ originating from point $C$ with angles $82^\circ$, $31^\circ$, and $g^\circ$ between them as shown in the diagram. 2. **Identify the angles:** The angle between $\overrightarrow{CE}$ and $\overrightarrow{AB}$ is $82^\circ$. The angle between $\overrightarrow{CD}$ and $\overrightarrow{AB}$ is $31^\circ$. The angle $g^\circ$ is between $\overrightarrow{CD}$ and $\overrightarrow{CE}$. 3. **Use the angle sum rule:** Since the vectors $\overrightarrow{AB}$, $\overrightarrow{CD}$, and $\overrightarrow{CE}$ are arranged around point $C$, the angles between them satisfy: $$ 31^\circ + g + 82^\circ = 180^\circ $$ This is because the three angles form a straight line around point $C$. 4. **Solve for $g$:** $$ g = 180^\circ - 31^\circ - 82^\circ $$ $$ g = 180^\circ - 113^\circ $$ $$ g = 67^\circ $$ 5. **Final answer:** $$ \boxed{67^\circ} $$ Thus, the value of $g$ is $67^\circ$.