Question

can the triangles be proven similar using the sss or sas similarity theorems?
○ yes, δefg ~ δklm only by sss.
○ yes, δefg ~ δklm only by sas.
○ yes, δefg ~ δklm by sss or sas.
○ no, they cannot be proven similar by sss or sas.
(diagrams: δefg with sides 24 (eg), 15 (fg), 18 (ef); δklm with sides 8 (km), 6 (kl), 5 (lm); angles marked at f (δefg) and l (δklm))

Explanation:

Step1: Check SAS Similarity

First, check the included angles (the marked angles) are equal. Then, find the ratios of the sides around the angles. For $\triangle EFG$ and $\triangle KLM$, the sides around the marked angles:

• In $\triangle EFG$: $EF = 18$, $FG = 15$
• In $\triangle KLM$: $LM = 5$, $KL = 6$

Calculate the ratios: $\frac{EF}{KL}=\frac{18}{6} = 3$, $\frac{FG}{LM}=\frac{15}{5}=3$. Since the included angles are equal (marked) and the sides around them are in proportion, SAS similarity holds.

Step2: Check SSS Similarity

Find all side ratios.

• $\triangle EFG$ sides: $EF = 18$, $FG = 15$, $EG = 24$
• $\triangle KLM$ sides: $KL = 6$, $LM = 5$, $KM = 8$

Ratios: $\frac{EF}{KL}=\frac{18}{6}=3$, $\frac{FG}{LM}=\frac{15}{5}=3$, $\frac{EG}{KM}=\frac{24}{8}=3$. All sides are in proportion, so SSS similarity holds.

Answer:

Yes, $\triangle EFG \sim \triangle KLM$ by SSS or SAS.