Question

determine if triangle opq and triangle rst are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Identify sides around shared angle

For $\triangle OPQ$: sides $\overline{OQ}=18$, $\overline{OP}=15$, included $\angle O=38^\circ$
For $\triangle RST$: sides $\overline{TR}=54$, $\overline{TS}=34$, included $\angle T$ (unknown)

Step2: Calculate side ratios

Ratio of $\frac{OQ}{TR} = \frac{18}{54} = \frac{1}{3}$
Ratio of $\frac{OP}{TS} = \frac{15}{34} \approx 0.441$

Step3: Compare ratios and angles

$\frac{1}{3}
eq \frac{15}{34}$, so the sides around the included angles are not proportional. There is no evidence the included angles are equal, and no other matching angle/side pairs satisfy similarity criteria (AA, SAS, SSS).

Answer:

The triangles are not similar.