Answer
Yes, the triangles are similar because two sides in one triangle are proportional to two sides in the other triangle, and the included angles in both triangles are congruent. Thus, $\triangle SPQ$ ~ $\triangle SRT$ using the SAS Similarity Theorem.
Work Step by Step
Let's set up the ratios of corresponding sides in $\triangle SPQ$ and $\triangle SRT$:
$\frac{SP}{SR} = \frac{24}{24 + 12}$
Add the denominator to simplify:
$\frac{SP}{SR} = \frac{24}{36}$
Divide both the numerator and denominator by their greatest common factor, $12$:
$\frac{SP}{SR} = \frac{2}{3}$
Let's look at $SQ$ and $ST$:
$\frac{SQ}{ST} = \frac{16}{16 + 8}$
Add the denominator to simplify:
$\frac{SQ}{ST} = \frac{16}{24}$
Divide both the numerator and denominator by their greatest common factor, $8$:
$\frac{SQ}{ST} = \frac{2}{3}$
The scale factor for the sides is congruent.
$\angle S$ is shared by both triangles, so the angle is a point of congruency in both triangles.
Yes, the triangles are similar because two sides in one triangle are proportional to two sides in the other triangle, and the included angles in both triangles are congruent. Thus, $\triangle SPQ$ ~ $\triangle SRT$ using the SAS Similarity Theorem.
