1. **State the problem:** We need to find the size of angle $x$ in a polygon with seven interior angles given as $108^\circ$, $121^\circ$, $62^\circ$, $274^\circ$ (reflex angle), $29^\circ$, $208^\circ$, and $x$. 2. **Recall the formula for the sum of interior angles of a polygon:** $$\text{Sum of interior angles} = (n - 2) \times 180^\circ$$ where $n$ is the number of sides. Here, $n=7$. 3. **Calculate the sum of interior angles:** $$ (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ $$ 4. **Note about the reflex angle:** The angle given as $274^\circ$ is a reflex interior angle, which means it is already the interior angle measure and should be used as is. 5. **Sum the known angles:** $$108^\circ + 121^\circ + 62^\circ + 274^\circ + 29^\circ + 208^\circ = 802^\circ$$ 6. **Set up the equation to find $x$:** $$802^\circ + x = 900^\circ$$ 7. **Solve for $x$:** $$x = 900^\circ - 802^\circ = 98^\circ$$ **Final answer:** $$\boxed{98^\circ}$$