1. **Problem statement:** Given an inscribed quadrilateral with angles labeled $5x$, $104^\circ$, $2x$, and $110^\circ$, find the value(s) of $x$. 2. **Formula and rule:** In a cyclic quadrilateral (one inscribed in a circle), opposite angles sum to $180^\circ$. 3. **Set up equations:** - Opposite angles $5x$ and $110^\circ$ satisfy: $$5x + 110 = 180$$ - Opposite angles $2x$ and $104^\circ$ satisfy: $$2x + 104 = 180$$ 4. **Solve first equation:** $$5x + 110 = 180$$ $$5x = 180 - 110$$ $$5x = 70$$ $$x = \frac{70}{5}$$ $$x = 14$$ 5. **Solve second equation:** $$2x + 104 = 180$$ $$2x = 180 - 104$$ $$2x = 76$$ $$x = \frac{76}{2}$$ $$x = 38$$ 6. **Check for consistency:** The two values for $x$ must be equal for the figure to be consistent. Since $14 \neq 38$, re-examine the labeling or problem context. Usually, the problem expects one $x$ value, so the correct pairs of opposite angles must be identified. 7. **Re-examining pairs:** - If $5x$ is opposite $104^\circ$, then: $$5x + 104 = 180$$ $$5x = 76$$ $$x = \frac{76}{5} = 15.2$$ - If $2x$ is opposite $110^\circ$, then: $$2x + 110 = 180$$ $$2x = 70$$ $$x = 35$$ 8. **Conclusion:** The problem likely pairs $5x$ with $104^\circ$ and $2x$ with $110^\circ$. The values of $x$ are $15.2$ and $35$, which are inconsistent. Since $x$ must be a single value, the problem likely has a typo or requires choosing the correct pair. **Final answer:** $x = 15.2$ if $5x$ is opposite $104^\circ$, or $x = 35$ if $2x$ is opposite $110^\circ$. This completes the solution for the first problem.