1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are $(100 - 2x)^\circ$ and $(10x)^\circ$. We need to solve for $x$. 2. **Identify the relationship:** Since the lines are parallel and the angles are on opposite sides of the transversal but inside the parallel lines, these angles are alternate interior angles and are 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 $x$ that satisfies the angle relationship is $x = \frac{25}{3}$ or approximately 8.33 degrees.