In the figure,if $\triangle ABC \sim \triangle DEF$ and their side lengths (in $cm$) are as marked,find the lengths of the sides of each triangle.

(N/A) Given that $\triangle ABC \sim \triangle DEF$.
Therefore,the ratios of their corresponding sides are equal:
$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$
Substituting the given expressions:
$\frac{2x - 1}{18} = \frac{2x + 2}{3x + 9} = \frac{3x}{6x}$
Taking the first and third ratios:
$\frac{2x - 1}{18} = \frac{3x}{6x}$
$\frac{2x - 1}{18} = \frac{1}{2}$
Cross-multiplying:
$2(2x - 1) = 18$
$4x - 2 = 18$
$4x = 20$
$x = 5$
Now,calculating the side lengths:
For $\triangle ABC$:
$AB = 2(5) - 1 = 9 \, cm$
$BC = 2(5) + 2 = 12 \, cm$
$CA = 3(5) = 15 \, cm$
For $\triangle DEF$:
$DE = 18 \, cm$
$EF = 3(5) + 9 = 24 \, cm$
$FD = 6(5) = 30 \, cm$