Question

find the scale factor of the figures. then list all pairs of congruent angles.
$defg \sim pqrs$
$k=\square$
$\angle d\cong\angle\square, \angle e\cong\angle\square, \angle f\cong\angle\square, \angle g\cong\angle\square$
write the ratios of the corresponding side lengths in a statement of proportionality.
$\square=\square=\square=\square$

Explanation:

Step1: Calculate scale factor

The scale factor $k$ is the ratio of corresponding sides of the smaller figure to the larger figure. Using $EF = 3$ and $PQ = 9$:
$k=\frac{PQ}{EF}=\frac{3}{9}=\frac{1}{3}$
(Verify with other sides: $\frac{QR}{FG}=\frac{1}{6}=\frac{1}{3}$, $\frac{RS}{GD}=\frac{4}{12}=\frac{1}{3}$, $\frac{SP}{DE}=\frac{2}{6}=\frac{1}{3}$)

Step2: Match congruent angles

For similar figures, corresponding angles are congruent. Using the similarity statement $DEFG \sim PQRS$ (mapping $D\to S, E\to P, F\to Q, G\to R$):
$\angle D\cong\angle S$, $\angle E\cong\angle P$, $\angle F\cong\angle Q$, $\angle G\cong\angle R$

Step3: Write proportionality statement

Set up ratios of corresponding sides of the smaller figure to the larger figure:
$\frac{DE}{SP}=\frac{EF}{PQ}=\frac{FG}{QR}=\frac{GD}{RS}=\frac{1}{3}$

Answer:

$k=\frac{1}{3}$
$\angle D\cong\angle S$, $\angle E\cong\angle P$, $\angle F\cong\angle Q$, $\angle G\cong\angle R$
$\frac{DE}{SP}=\frac{EF}{PQ}=\frac{FG}{QR}=\frac{GD}{RS}=\frac{1}{3}$