Question

are the triangles similar? if so, what postulate or theorem proves their similarity?
no, the triangles are not similar.
yes, by aa similarity postulate
yes, by sas similarity theorem
yes, by sss similarity theorem

Explanation:

Step1: Check side - length ratios

Find the ratios of corresponding sides. For $\triangle ABC$ with sides $a = 10$, $b = 11$, $c = 12$ and $\triangle DEF$ with sides $d = 35$, $e = 38.5$, $f = 42$. Calculate $\frac{d}{a}=\frac{35}{10}=3.5$, $\frac{e}{b}=\frac{38.5}{11}=3.5$, $\frac{f}{c}=\frac{42}{12}=3.5$.

Step2: Apply similarity theorem

Since the ratios of the corresponding sides of the two triangles are equal ($\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} = 3.5$), by the SSS (Side - Side - Side) Similarity Theorem, the two triangles are similar.

Answer:

Yes, by SSS Similarity Theorem