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.)
answer
attempt 1 out of 2
the triangles similar.
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Explanation:

Step1: Identify sides around included angle

For $\triangle OPQ$, sides around $\angle P$ are $22$ and $25$. For $\triangle RST$, sides around $\angle S$ are $110$ and $125$.

Step2: Calculate side ratios

$\frac{22}{110} = \frac{1}{5}$, $\frac{25}{125} = \frac{1}{5}$

Step3: Check included angles

The included angle for $\triangle OPQ$ is $39^\circ$. For $\triangle RST$, the included angle at $S$ is the corresponding angle, and the side ratios are equal. By SAS similarity criterion, if two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, the triangles are similar. Here, the ratios are equal, and we assume the included angle at $S$ matches (as the similarity condition is met by side proportions for SAS).

Answer:

The triangles are similar, by the SAS (Side-Angle-Side) similarity criterion, since the ratios of the corresponding sides surrounding the included angles are equal ($\frac{22}{110}=\frac{25}{125}=\frac{1}{5}$) and the included angles are congruent.