The problem asks if two quadrilaterals, $ABCD$ and $NKLM$, are similar. If they are similar, we need to write a similarity statement and find the scale factor.

1. Problem Description

The problem asks if two quadrilaterals, $ABCD$ and $NKLM$, are similar. If they are similar, we need to write a similarity statement and find the scale factor.

2. Solution Steps

To determine if the two quadrilaterals are similar, we need to check if their corresponding angles are congruent and if their corresponding sides are proportional.

The given angle at $A$ is $120^\circ$, and the given angle at $M$ is $120^\circ$. Assuming that these angles correspond, we have $\angle A \cong \angle M$.

Assuming that all the other corresponding angles are also congruent (although not explicitly given), we can move on to checking the proportionality of corresponding sides.

We have $AB = 10$, $AD = 5$, $DC = 10$, and $BC = 5$.

Also, we have $NK = 1.2$, $NM = 3.12$, $ML = 1.2$, and $KL = 3.12$.

Let's check the ratios of the sides:

$\frac{AB}{ML} = \frac{10}{1.2} = \frac{100}{12} = \frac{25}{3}$

$\frac{AD}{NM} = \frac{5}{3.12} = \frac{500}{312} = \frac{125}{78}$

$\frac{BC}{NK} = \frac{5}{1.2} = \frac{50}{12} = \frac{25}{6}$

$\frac{DC}{KL} = \frac{10}{3.12} = \frac{1000}{312} = \frac{125}{39}$

Since the ratios of corresponding sides are not equal, the quadrilaterals are not similar. For instance $\frac{AB}{ML} = \frac{25}{3}$ and $\frac{AD}{NM} = \frac{125}{78}$. Also $\frac{AD}{NM}$ is not equal to $\frac{BC}{NK}$. Therefore, the polygons are not similar.

If instead, we assume that $AB$ corresponds to $NK$, and $AD$ corresponds to $NM$, then

$\frac{AB}{NK} = \frac{10}{1.2} = \frac{100}{12} = \frac{25}{3}$.

$\frac{AD}{NM} = \frac{5}{3.12} = \frac{500}{312} = \frac{125}{78} = \frac{25}{3} \times \frac{5}{26}$

Therefore, the ratio between $AD$ and $NM$ is $\frac{5}{26}$ of the ratio of $AB$ to $NK$. Thus, they are not similar.

3. Final Answer

The polygons are not similar.