1. **Problem Statement:** Given three sets of parallel lines: $a \parallel b$, $c \parallel d$, and $e \parallel f$, and the segments with lengths 12, 5, and 8, find the missing side lengths $x$, $y$, and $z$. 2. **Key Concept:** When lines are parallel, corresponding segments between these lines are proportional. This means the ratios of the lengths of segments between parallel lines are equal. 3. **Set up the proportions:** Since $a \parallel b$, $c \parallel d$, and $e \parallel f$, the segments between these lines satisfy: $$\frac{x}{12} = \frac{y}{5} = \frac{z}{8}$$ 4. **Find $x$, $y$, and $z$:** Let the common ratio be $k$, so: $$x = 12k, \quad y = 5k, \quad z = 8k$$ 5. **Use the fact that the sum of segments equals the total length:** Assuming the total length along the line is the sum of the segments, we have: $$x + y + z = 12 + 5 + 8 = 25$$ Substitute the expressions in terms of $k$: $$12k + 5k + 8k = 25$$ $$25k = 25$$ 6. **Solve for $k$:** $$k = \frac{25}{25} = 1$$ 7. **Calculate the missing lengths:** $$x = 12 \times 1 = 12$$ $$y = 5 \times 1 = 5$$ $$z = 8 \times 1 = 8$$ **Final answer:** $$x = 12, \quad y = 5, \quad z = 8$$