To solve this problem using the SSS (Side-Side-Side) similarity theorem, let's break down the steps involved:
1. Understand the SSS Similarity Theorem:
- The SSS similarity theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
2. Identify Corresponding Side Ratios:
- Given two similar triangles, [tex]\(\triangle HLI\)[/tex] and [tex]\(\triangle JLK\)[/tex], their corresponding sides must be proportional. This means:
[tex]\[
\frac{HL}{JL} = \frac{IL}{KL}
\][/tex]
3. Determine the Corresponding Ratio:
- To find which other ratio is equal to [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex], we need to look at the remaining pair of corresponding sides from the two triangles. When two triangles are similar, all corresponding side lengths are proportional.
- Along with the sides [tex]\(HL\)[/tex] and [tex]\(JL\)[/tex], and [tex]\(IL\)[/tex] and [tex]\(KL\)[/tex], the corresponding third sides would be [tex]\(HI\)[/tex] from [tex]\(\triangle HLI\)[/tex] and [tex]\(JK\)[/tex] from [tex]\(\triangle JLK\)[/tex]. Hence:
[tex]\[
\frac{HL}{JL} = \frac{IL}{KL} = \frac{HI}{JK}
\][/tex]
4. Conclusion:
- From the steps above, we can conclude that:
[tex]\[
\frac{HL}{JL} = \frac{IL}{KL} = \frac{HI}{JK}
\][/tex]
Therefore, [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex] is also equal to the ratio [tex]\(\frac{HI}{JK}\)[/tex].
Thus, the correct answer is:
[tex]\[
\frac{HI}{JK}
\][/tex]
