1. **State the problem:** We have three angles around a point with measures $(3x + 8)^\circ$, $(1gy + 4 \cdot 10)^\circ$, and $(9x - 22)^\circ$. We need to find the values of $x$ and $y$. 2. **Understand the geometry rule:** The sum of angles around a point is always $360^\circ$. 3. **Write the equation:** $$ (3x + 8) + (1gy + 4 \cdot 10) + (9x - 22) = 360 $$ 4. **Simplify the terms:** Calculate $4 \cdot 10 = 40$. So the equation becomes: $$ 3x + 8 + 1gy + 40 + 9x - 22 = 360 $$ 5. **Combine like terms:** $$ (3x + 9x) + (8 + 40 - 22) + 1gy = 360 $$ $$ 12x + 26 + 1gy = 360 $$ 6. **Isolate $1gy$:** $$ 1gy = 360 - 12x - 26 $$ $$ 1gy = 334 - 12x $$ 7. **Interpret $1gy$:** Since $1gy$ is likely a typo or variable, assume it means $y$ (or $1 \times g \times y$). If $g=1$, then $y = 334 - 12x$. 8. **Find $x$ and $y$ values:** Without additional information, we cannot find unique values for $x$ and $y$. If the problem expects $y$ to be independent, more data is needed. **Final answer:** $$ y = 334 - 12x $$ Since no further info is given, $x$ remains a variable.