1. **Problem Statement:** Given two parallel lines $m \parallel n$ and a transversal $t$, find the value of $x$ given the angles $(x - 30)^\circ$ and $(3x - 10)^\circ$ at the intersection with line $m$. 2. **Key Concept:** When a transversal intersects two parallel lines, alternate interior angles are equal, and adjacent angles on a straight line sum to $180^\circ$. 3. **Identify the relationship:** The angles $(x - 30)^\circ$ (upper left) and $(3x - 10)^\circ$ (lower right) at the same intersection are **vertical angles**, which are always equal. 4. **Set up the equation:** $$ x - 30 = 3x - 10 $$ 5. **Solve for $x$:** $$ x - 30 = 3x - 10 \\ -30 + 10 = 3x - x \\ -20 = 2x \\ x = \frac{-20}{2} = -10 $$ 6. **Interpretation:** The value of $x$ is $-10$. This satisfies the equality of vertical angles at the intersection of the transversal with line $m$. **Final answer:** $x = -10$