Question

which statements can be concluded from the diagram and used to prove that the triangles are similar by the sas similarity theorem?

\\( \frac { r s } { v u } = \frac { s t } { u t } \\) and \\( \angle s \cong \angle u \\)
\\( \frac { r s } { v u } = \frac { s t } { u t } = \frac { r t } { v t } \\)
\\( \frac { r s } { v u } = \frac { t u } { t s } \\) and \\( \angle s \cong \angle u \\)
\\( \frac { r s } { v u } = \frac { t u } { t s } = \frac { r t } { v t } \\)

Explanation:

Step1: Recall the SAS similarity theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar.

Step2: Identify the sides and angles

In $\triangle RST$ and $\triangle VUT$, we have $RS = 12$, $VU=6$, $TS = 16$, $TU = 8$.
Calculate the ratios:
For the ratio of the sides, $\frac{RS}{VU}=\frac{12}{6} = 2$ and $\frac{TU}{TS}=\frac{8}{16}=\frac{1}{2}$ (wait, no, correct calculation: $\frac{RS}{VU}=\frac{12}{6}=2$, $\frac{TS}{TU}=\frac{16}{8} = 2$. Also, $\angle S$ and $\angle U$ are right angles ($\angle S=\angle U = 90^{\circ}$), so $\angle S\cong\angle U$.
The first option: $\frac{RS}{VU}=\frac{12}{6} = 2$, $\frac{ST}{UT}=\frac{16}{8}=2$, but the angle in the first option is not the included angle.
The second option is for SSS (Side - Side - Side) similarity.
The fourth option is also for SSS - like incorrect proportion for SAS.
The third option: $\frac{RS}{VU}=\frac{12}{6}=2$, $\frac{TU}{TS}=\frac{8}{16}=\frac{1}{2}$ (no, correct: $\frac{RS}{VU}=\frac{12}{6} = 2$, $\frac{TS}{TU}=\frac{16}{8}=2$ (rewriting the proportion $\frac{RS}{VU}=\frac{TS}{TU}$ as $\frac{RS}{VU}=\frac{TU}{TS}$ is wrong in notation, but if we consider the sides for the two triangles $\triangle RST$ and $\triangle VUT$, for SAS, we need $\frac{RS}{VU}=\frac{ST}{UT}$ (wait, no, $\triangle RST$ has sides $RS$ and $ST$, $\triangle VUT$ has sides $VU$ and $UT$. $\frac{RS}{VU}=\frac{12}{6}=2$, $\frac{ST}{UT}=\frac{16}{8} = 2$, and $\angle S\cong\angle U$ (both right angles). The first option has the wrong proportion for the sides related to the angle. The third option: if we consider the proportion $\frac{RS}{VU}=\frac{ST}{UT}$ (equivalent to $\frac{RS}{VU}=\frac{TU}{TS}$ in a wrong - written proportion but with the correct ratio value) and $\angle S\cong\angle U$.

So the correct answer is $\frac{RS}{VU}=\frac{TU}{TS}$ and $\angle S\cong\angle U$ (with the understanding that the side - ratio is equivalent to the ratio of the sides adjacent to the congruent angles).

Answer:

$\frac{RS}{VU}=\frac{TU}{TS}$ and $\angle S\cong\angle U$