Answer
Yes, the two triangles are similar because corresponding sides have the same scale factor, meaning the sides are proportional, and the included angle is congruent. $\triangle KNP$ ~ $\triangle DGK$ by the SAS Similarity Theorem.
Work Step by Step
We are given the measures of two sides and an included angle in the two triangles. Let's see if the Side-Angle-Side Similarity Theorem can be applied here.
The SAS Similarity Theorem states that if two sides in one triangle are proportional to the two sides of another triangle, and the included angle in one triangle is congruent to the included angle of the other triangle, then the two triangles are similar.
Let's set up the ratios of corresponding sides in $\triangle KNP$ and $\triangle DGK$:
$\frac{NP}{GK} = \frac{16}{12}$
Divide both the numerator and denominator by their greatest common factor, $4$:
$\frac{NP}{GK} = \frac{4}{3}$
Let's look at $KN$ and $DG$:
$\frac{KN}{DG} = \frac{12}{9}$
Divide both the numerator and denominator by their greatest common factor, $3$:
$\frac{KN}{DG} = \frac{4}{3}$
Both $\angle N$ and $\angle G$ are right angles; therefore, these two angles are congruent.
So, yes, the two triangles are similar because corresponding sides have the same scale factor. $\triangle KNP$ ~ $\triangle DGK$ by the SAS Similarity Theorem.
