1. The problem involves two intersecting lines forming vertical angles. We are given one angle as $102^\circ$ and the opposite angle as $(10x - 2)^\circ$. 2. Vertical angles are always equal, so we set the expressions equal: $$102 = 10x - 2$$ 3. Add 2 to both sides to isolate the term with $x$: $$102 + 2 = 10x$$ $$104 = 10x$$ 4. Divide both sides by 10 to solve for $x$: $$x = \frac{104}{10} = 10.4$$ Therefore, the value of $x$ is $10.4$.