To solve for the measure of side $HI$, we can use the fact that triangles $\triangle DEF$ and $\triangle GHI$ are similar. This means their corresponding sides are proportional.

Let's break it down:

Given the corresponding sides of two similar triangles, we have the proportion:

$\frac{DE}{GH} = \frac{EF}{HI} = \frac{DF}{GI}$

Now, we need more specific information, such as the lengths of the sides or some of the angles in both triangles, to set up the correct ratio and solve for $HI$. It seems like you might have forgotten to mention the side lengths of either triangle.

Could you clarify the side lengths of triangle DEF and the corresponding sides in triangle GHI?