1. **Problem statement:** Given angle $FGC = 98^\circ$ and arc $FC = 60^\circ$, find the measure of arc $FB$. 2. **Understanding the problem:** In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. 3. **Formula:** For an inscribed angle $\angle FGC$, the measure is $$\angle FGC = \frac{1}{2} \times \text{measure of arc } FC + \text{measure of arc } FB.$$ Since $\angle FGC$ intercepts arcs $FC$ and $FB$, we have $$98 = \frac{1}{2} (\text{arc } FC + \text{arc } FB).$$ 4. **Substitute known values:** $$98 = \frac{1}{2} (60 + \text{arc } FB).$$ 5. **Solve for arc $FB$:** Multiply both sides by 2: $$2 \times 98 = 60 + \text{arc } FB$$ $$196 = 60 + \text{arc } FB$$ Subtract 60 from both sides: $$\cancel{196} - \cancel{60} = \text{arc } FB$$ $$136 = \text{arc } FB$$ 6. **Check the circle's total arc:** The total circle is $360^\circ$. Since arc $FC = 60^\circ$ and arc $FB = 136^\circ$, the remaining arcs must sum to $360 - 60 - 136 = 164^\circ$. 7. **Conclusion:** The measure of arc $FB$ is $136^\circ$. However, this value is not among the choices given. **Re-examining the problem:** The angle $FGC$ is an exterior angle to triangle $FGC$ and intercepts arcs $FB$ and $FC$ differently. The exterior angle theorem for circles states: $$\angle FGC = \frac{1}{2} |\text{arc } FB - \text{arc } FC|.$$ Using this: $$98 = \frac{1}{2} |\text{arc } FB - 60|$$ Multiply both sides by 2: $$196 = |\text{arc } FB - 60|$$ Since arc measures cannot exceed 360, the only possible solution is: $$\text{arc } FB - 60 = -196$$ $$\text{arc } FB = 60 - 196 = -136$$ (not possible) Or $$\text{arc } FB - 60 = 196$$ $$\text{arc } FB = 256$$ (too large for the given choices) Since these do not match choices, consider the smaller arc between points $F$ and $B$ on the circle, which is $360 - 256 = 104$ (not in choices). **Alternative approach:** The angle $FGC$ is an inscribed angle intercepting arc $FB$ only, so: $$\angle FGC = \frac{1}{2} \times \text{arc } FB$$ Then: $$98 = \frac{1}{2} \times \text{arc } FB$$ $$\text{arc } FB = 196$$ (not in choices) **Final step:** The problem likely expects the difference between arcs $FB$ and $FC$ to be $2 \times 98 = 196$, so: $$\text{arc } FB = 196 - 60 = 136$$ (not in choices) Given the choices, the closest reasonable answer is $48^\circ$ (which is $60 - 12$ or $60 - 12$), but this does not fit the calculations. **Answer:** None of the given choices match the correct calculation of arc $FB = 136^\circ$. **However, if the angle $FGC$ is an inscribed angle intercepting arc $FB$, then:** $$\angle FGC = \frac{1}{2} \times \text{arc } FB$$ $$98 = \frac{1}{2} \times \text{arc } FB$$ $$\text{arc } FB = 196$$ Since $196$ is not an option, the problem might have a typo or missing information. **Therefore, based on the standard circle theorems, the measure of arc $FB$ is $136^\circ$.**