Question

1. which similarity theorem, if any is used to prove these triangles similar

Explanation:

Step1: Calculate the ratios of corresponding sides

For triangle \(FGH\) and \(KML\), we have the following pairs of corresponding sides:

• \(\frac{GH}{KL}=\frac{55}{143}=\frac{5}{13}\)
• \(\frac{FH}{ML}=\frac{40}{88}=\frac{5}{11}\)
• \(\frac{FG}{KM}=\frac{65}{121}\)

Since \(\frac{5}{13}
eq\frac{5}{11}
eq\frac{65}{121}\), we made a mistake. Let's re - check the correspondence.

If we assume \(FG\) corresponds to \(KL\), \(GH\) corresponds to \(ML\), \(FH\) corresponds to \(KM\)

• \(\frac{FG}{KL}=\frac{65}{143}=\frac{5}{11}\)
• \(\frac{GH}{ML}=\frac{55}{121}=\frac{5}{11}\)
• \(\frac{FH}{KM}=\frac{40}{88}=\frac{5}{11}\)

Step2: Apply the Side - Side - Side (SSS) similarity theorem

The Side - Side - Side (SSS) similarity theorem states that if the ratios of the lengths of the corresponding sides of two triangles are equal, then the two triangles are similar.

Since \(\frac{FG}{KL}=\frac{GH}{ML}=\frac{FH}{KM}=\frac{5}{11}\), by the SSS similarity theorem, \(\triangle FGH\sim\triangle KLM\)

Answer:

Side Side Side ~ (SSS similarity theorem is used)