1. **State the problem:** We have two parallel lines cut by a transversal, creating angles at points M and N. 2. **Identify the angles:** The angle at M is $(3x + 10)^\circ$ and the angle at N is $(5x - 30)^\circ$. 3. **Use the property of alternate interior angles:** Since the lines are parallel, the alternate interior angles are equal. 4. **Set up the equation:** $$3x + 10 = 5x - 30$$ 5. **Solve for $x$:** Subtract $3x$ from both sides: $$10 = 2x - 30$$ Add $30$ to both sides: $$40 = 2x$$ Divide both sides by $2$: $$x = 20$$ 6. **Conclusion:** The value of $x$ is $20$.