1. **State the problem:** Given two parallel lines $m \parallel n$ cut by a transversal, find the values of $x$ and $y$ given the angles $(2x + 10)^\circ$ on line $m$, and $(8x - 20)^\circ$ and $y^\circ$ on line $n$. 2. **Identify angle relationships:** Since $m \parallel n$, corresponding angles are equal, and alternate interior angles are equal. The angle $(2x + 10)^\circ$ on line $m$ corresponds to the angle $(8x - 20)^\circ$ on line $n$ because they are alternate interior angles. 3. **Set up the equation for $x$ using alternate interior angles:** $$ 2x + 10 = 8x - 20 $$ 4. **Solve for $x$:** $$ 2x + 10 = 8x - 20 \\ 10 + \cancel{2x} = 8x - 20 - \cancel{2x} \\ 10 = 6x - 20 \\ 10 + 20 = 6x \\ 30 = 6x \\ x = \frac{30}{6} = 5 $$ 5. **Find $y$:** The angle $y^\circ$ is adjacent to $(8x - 20)^\circ$ on line $n$ and together they form a straight line, so they are supplementary angles. 6. **Write the supplementary angle equation:** $$ (8x - 20) + y = 180 $$ 7. **Substitute $x=5$ into the equation:** $$ 8(5) - 20 + y = 180 \\ 40 - 20 + y = 180 \\ 20 + y = 180 $$ 8. **Solve for $y$:** $$ y = 180 - 20 = 160 $$ **Final answers:** $$ x = 5, \quad y = 160$$