1. **State the problem:** We have two parallel lines $m$ and $n$ cut by a transversal, creating angles labeled as $(9x+2)^\circ$, $(5x-18)^\circ$ near line $m$, and $(2y+20)^\circ$ near line $n$. We need to find the values of $x$ and $y$. 2. **Identify angle relationships:** Since $m$ and $n$ are parallel, corresponding angles are equal, and alternate interior angles are equal. 3. **Set up equations:** - The angles $(9x+2)^\circ$ and $(5x-18)^\circ$ are adjacent on a straight line, so they are supplementary: $$ (9x+2) + (5x-18) = 180 $$ - The angle $(9x+2)^\circ$ corresponds to $(2y+20)^\circ$ (corresponding angles), so: $$ 9x + 2 = 2y + 20 $$ 4. **Solve the first equation:** $$ 9x + 2 + 5x - 18 = 180 $$ $$ 14x - 16 = 180 $$ $$ 14x = 196 $$ $$ x = \frac{196}{14} $$ $$ x = 14 $$ 5. **Substitute $x=14$ into the second equation:** $$ 9(14) + 2 = 2y + 20 $$ $$ 126 + 2 = 2y + 20 $$ $$ 128 = 2y + 20 $$ $$ 2y = 128 - 20 $$ $$ 2y = 108 $$ $$ y = \frac{108}{2} $$ $$ y = 54 $$ 6. **Final answer:** $$ x = 14, \quad y = 54 $$