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 $square$ similar.

Explanation:

Step1: Identify sides around included angle

For $\triangle OPQ$, sides around $\angle P$: $22, 25$; included angle $\angle P=39^\circ$.
For $\triangle RST$, sides around $\angle S$: $110, 125$.

Step2: Calculate side ratios

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

Step3: Check SAS similarity rule

The included angles are not given as equal, but first confirm if the angle at $S$ matches? Wait, no: Wait, we check if the sides around the included angle are proportional AND the included angles are equal. Wait, no, we first check the ratios: $\frac{OP}{SR}=\frac{22}{110}=\frac{1}{5}$, $\frac{PQ}{ST}=\frac{25}{125}=\frac{1}{5}$, and if $\angle P = \angle S$? Wait no, wait the sides: $\triangle OPQ$: $OP=22$, $PQ=25$, $OQ=16$; $\triangle RST$: $SR=110$, $ST=125$, $RT=80$.
Wait another way: Check all side ratios:
$\frac{OQ}{RT}=\frac{16}{80}=\frac{1}{5}$, $\frac{OP}{SR}=\frac{22}{110}=\frac{1}{5}$, $\frac{PQ}{ST}=\frac{25}{125}=\frac{1}{5}$

Step4: Apply SSS similarity rule

All corresponding sides have equal ratios.

Answer:

The triangles are similar, by the SSS (Side-Side-Side) similarity criterion, as all pairs of corresponding sides are in the same ratio of $\frac{1}{5}$.