1. **State the problem:** We are given two angles at point O, one measuring $7x$ and the other $8x$, and the angle between these two rays is $60^\circ$. We need to find the value of $x$. 2. **Understand the geometry:** The rays B-O and C-O form two adjacent angles with the horizontal line A-O-D. The sum of these two angles minus the $60^\circ$ angle between them equals the straight angle $180^\circ$ because A-O-D is a straight line. 3. **Set up the equation:** The sum of the angles around point O on a straight line is $180^\circ$. The two angles $7x$ and $8x$ overlap by $60^\circ$, so: $$7x + 8x - 60 = 180$$ 4. **Simplify the equation:** $$15x - 60 = 180$$ 5. **Solve for $x$:** $$15x = 180 + 60$$ $$15x = 240$$ $$x = \frac{240}{15}$$ $$x = 16$$ 6. **Conclusion:** The value of $x$ is $16$. **Final answer:** $x = 16$