1. The problem involves three parallel lines $\overline{BC} \parallel \overline{DE} \parallel \overline{FG}$ and points on these lines forming segments. We need to complete each proportion by identifying the missing segment in the ratio. 2. Important rule: When three or more parallel lines intersect two transversals, the segments they cut on the transversals are proportional. This means: $$\frac{AB}{BD} = \frac{AC}{CE} = \frac{AF}{FG}$$ 3. Using this rule, we complete each proportion: 15. Given $\frac{AB}{BD} = \frac{AC}{\square}$, the missing segment is $CE$ because $AC$ corresponds to the segment on the other transversal matching $BD$. 16. Given $\frac{\square}{DF} = \frac{AE}{EG}$, the missing segment is $BD$ because $BD$ corresponds to $AE$ on the other transversal. 17. Given $\frac{DF}{\square} = \frac{EG}{CE}$, the missing segment is $FG$ because $FG$ corresponds to $CE$. 18. Given $\frac{AF}{AB} = \frac{\square}{AC}$, the missing segment is $BF$ because $BF$ corresponds to $AB$. 19. Given $\frac{BD}{CE} = \frac{\square}{EG}$, the missing segment is $AF$ because $AF$ corresponds to $BD$. 20. Given $\frac{AB}{AC} = \frac{BF}{\square}$, the missing segment is $FG$ because $FG$ corresponds to $AC$. Final answers: 15. $CE$ 16. $BD$ 17. $FG$ 18. $BF$ 19. $AF$ 20. $FG$