1. **State the problem:** We need to find the measure of angle $\angle FGJ$. 2. **Given information:** The angle between $FG$ and $GH$ is $57^\circ$, and the angle between $GH$ and $GJ$ is $41^\circ$. 3. **Formula and rule:** The measure of two adjacent angles can be added to find the total measure of the larger angle they form. So, $$m\angle FGJ = m\angle FGH + m\angle GHJ$$ 4. **Substitute the values:** $$m\angle FGJ = 57^\circ + 41^\circ$$ 5. **Calculate the sum:** $$m\angle FGJ = 98^\circ$$ 6. **Conclusion:** The measure of $\angle FGJ$ is $98^\circ$.