1. **Stating the problem:** We have three angles related by the diagram: one angle is $5y$, another is $4y$, and the angle between these two rays is $y$. We want to find the value of $y$. 2. **Understanding the problem:** The three rays start from the same point on a horizontal line. The green ray forms an angle $5y$ with the horizontal line to the left, the purple ray forms an angle $4y$ with the horizontal line to the right, and the red arc between the green and purple rays is labeled $y$. 3. **Key insight:** Since the green ray is $5y$ degrees above the horizontal line to the left and the purple ray is $4y$ degrees above the horizontal line to the right, the total angle between the green and purple rays is the sum of these two angles: $$\text{Angle between rays} = 5y + 4y = 9y$$ 4. **Given that the angle between the rays is also labeled $y$, we set up the equation:** $$9y = y$$ 5. **Solving the equation:** Subtract $y$ from both sides: $$9y - y = 0 \implies 8y = 0$$ Divide both sides by 8: $$y = 0$$ 6. **Interpretation:** The only solution is $y=0$, which means the angles collapse to zero. This suggests the labeling might represent a different relationship or the problem is to find $y$ under these constraints. 7. **Alternative interpretation:** If the angle between the rays is $y$, and the rays form angles $5y$ and $4y$ with the horizontal line on opposite sides, then the total angle between the rays is $5y + 4y = 9y$, which should equal $y$ (the angle between them). This is only possible if $y=0$. **Final answer:** $$y = 0$$