1. **State the problem:** We have two parallel lines $m$ and $n$ cut by a transversal. On line $m$, two adjacent angles are given as $(3x - 20)^\circ$ and $(y - 6)^\circ$. On line $n$, the angle at the intersection with the transversal is $(2x + 3)^\circ$. We need to find the values of $x$ and $y$. 2. **Use angle rules:** Adjacent angles on a straight line sum to $180^\circ$. Since $(3x - 20)^\circ$ and $(y - 6)^\circ$ are adjacent on line $m$, we have: $$ (3x - 20) + (y - 6) = 180 $$ Simplify: $$ 3x - 20 + y - 6 = 180 $$ $$ 3x + y - 26 = 180 $$ $$ 3x + y = 206 \quad \text{(Equation 1)} $$ 3. **Use alternate interior angles:** Because lines $m$ and $n$ are parallel, alternate interior angles are equal. The angle $(3x - 20)^\circ$ on line $m$ corresponds to the angle $(2x + 3)^\circ$ on line $n$: $$ 3x - 20 = 2x + 3 $$ Subtract $2x$ from both sides: $$ 3x - 2x - 20 = 3 $$ $$ x - 20 = 3 $$ Add $20$ to both sides: $$ x = 23 $$ 4. **Find $y$:** Substitute $x=23$ into Equation 1: $$ 3(23) + y = 206 $$ $$ 69 + y = 206 $$ Subtract $69$ from both sides: $$ y = 206 - 69 $$ $$ y = 137 $$ **Final answers:** $$ x = 23, \quad y = 137 $$