To find the length of \( LQ \) in the triangles \( \triangle JKL \) and \( \triangle PKQ \), we can use the property of similar triangles. Since \( \triangle JKL \sim \triangle PKQ \), the ratios of their corresponding sides will be equal.
Given:
- \( JL = 15 \) inches
- \( KQ = 17 \) inches
- \( PK = 15 \) inches
Let's denote \( LQ \) as \( x \).
Using the similarity of the triangles, we can set up the following proportion based on the corresponding sides:
\[ \frac{JL}{PK} = \frac{LQ}{KQ} \]
Substituting in the values we know:
\[ \frac{15}{15} = \frac{x}{17} \]
This simplifies to:
\[ 1 = \frac{x}{17} \]
Now we can solve for \( x \):
\[ x = 17 \]
Thus, the length of \( LQ \) is:
\[ \boxed{17} \text{ inches} \]