1. **State the problem:** We are given two parallel lines cut by a transversal, creating segments of lengths 14 and 8 on one side, and 12 and an unknown length $x$ on the other side. We need to find the value of $x$. 2. **Use the property of parallel lines cut by a transversal:** The segments on one side are proportional to the segments on the other side. This means: $$\frac{14}{8} = \frac{12}{x}$$ 3. **Set up the equation:** $$\frac{14}{8} = \frac{12}{x}$$ 4. **Cross multiply to solve for $x$:** $$14 \times x = 8 \times 12$$ 5. **Calculate the right side:** $$14x = 96$$ 6. **Divide both sides by 14 to isolate $x$:** $$x = \frac{96}{14}$$ 7. **Simplify the fraction by canceling common factors:** $$x = \frac{\cancel{96}^{48}}{\cancel{14}^{7}} = \frac{48}{7}$$ 8. **Convert to decimal (optional):** $$x \approx 6.857$$ **Final answer:** $$x = \frac{48}{7} \approx 6.857$$