Question

can the triangles be proven similar using the sss or sas similarity theorems?
○ yes, △efg ~ △klm only by sss.
○ yes, △efg ~ △klm only by sas.
○ yes, △efg ~ △klm by sss or sas.
○ no, they cannot be proven similar by sss or sas.

Explanation:

Step1: Check SAS Similarity

First, check the included angles. The marked angles (at F and L) are equal (given by the red arcs). Now, find the ratios of the sides around the equal angles. For $\triangle EFG$ and $\triangle KLM$, the sides around the equal angles:

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

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

Step2: Check SSS Similarity

Find all side ratios.

• $EF = 18$, $KL = 6$: $\frac{18}{6} = 3$
• $FG = 15$, $LM = 5$: $\frac{15}{5} = 3$
• $EG = 24$, $KM = 8$: $\frac{24}{8} = 3$

All three sides are in proportion ($3$), so SSS similarity also holds.

Answer:

Yes, $\triangle EFG \sim \triangle KLM$ by SSS or SAS. (Corresponding option: "Yes, $\triangle EFG \sim \triangle KLM$ by SSS or SAS.")