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.)
answer
the triangles
are
are not
similar.

Explanation:

Step1: Identify sides around included angle

For $\triangle DEF$, sides $\overline{EF}=17$, $\overline{ED}=21$, included $\angle E=38^\circ$.
For $\triangle GHI$, match sides to check proportionality:
Calculate ratios of corresponding sides:
$\frac{EF}{IG}=\frac{17}{52}$, $\frac{ED}{GH}=\frac{21}{84}=\frac{1}{4}$

Step2: Check SAS similarity condition

SAS similarity requires:

1. Included angles are equal
2. Ratios of surrounding sides are equal

Here, $\frac{17}{52}
eq \frac{1}{4}$, so side ratios are not equal. No included angle is given for $\triangle GHI$ to match $\angle E$, and the side ratios do not align for SAS. Checking all side ratios:
$\frac{EF}{IG}=\frac{17}{52}\approx0.327$, $\frac{ED}{GH}=\frac{21}{84}=0.25$, $\frac{FD}{HI}=\frac{13}{70}\approx0.186$
All ratios are different, so SSS similarity also fails.

Answer:

The triangles are not similar.