1. **State the problem:** Given two parallel lines $m \parallel n$ cut by a transversal, find the value of $x$ when one angle is $85^\circ$ and the other angle is $(3x + 13)^\circ$ on the same side of the transversal. 2. **Identify the relationship:** When two parallel lines are cut by a transversal, the consecutive interior angles on the same side of the transversal are supplementary. This means their measures add up to $180^\circ$. 3. **Set up the equation:** $$85 + (3x + 13) = 180$$ 4. **Simplify the equation:** $$85 + 3x + 13 = 180$$ $$3x + 98 = 180$$ 5. **Isolate $x$:** $$3x = 180 - 98$$ $$3x = 82$$ 6. **Solve for $x$:** $$x = \frac{82}{3}$$ 7. **Show cancellation step:** $$x = \cancel{\frac{82}{3}}$$ (no common factors to cancel further) 8. **Final answer:** $$x = \frac{82}{3} \approx 27.33$$ This means the value of $x$ is $\frac{82}{3}$ or approximately $27.33$.