1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are $(100 - 2x)^\circ$ at point K and $(10x)^\circ$ at point L. We need to solve for $x$. 2. **Identify the relationship:** Since the lines are parallel and the transversal intersects them, the angles at K and L are corresponding angles and therefore equal. 3. **Set up the equation:** $$100 - 2x = 10x$$ 4. **Solve for $x$:** \begin{align*} 100 - 2x &= 10x \\ 100 &= 10x + 2x \\ 100 &= 12x \\ x &= \frac{100}{12} = \frac{25}{3} \approx 8.33 \end{align*} 5. **Conclusion:** The value of $x$ is $\frac{25}{3}$ or approximately 8.33 degrees.