Question

determine if triangle def and triangle ghi are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
triangle def has sides: df = 13, ef = 17, de = 21, and angle at e is 38 degrees.
triangle ghi has sides: gi = 52, hi = 84, gh = 70.

Explanation:

Step1: Identify side pairs around angle

For $\triangle DEF$, sides around $\angle E$: $FE=17$, $DE=21$
For $\triangle GHI$, sides around the marked angle: $GH=84$, $HI=70$

Step2: Calculate side ratios

Ratio of $FE$ to $GH$: $\frac{17}{84}$
Ratio of $DE$ to $HI$: $\frac{21}{70}=\frac{3}{10}=0.3$

Step3: Compare ratios and angles

$\frac{17}{84}\approx0.202
eq0.3$, so the side pairs around the included angles are not proportional.

Answer:

The triangles are not similar.