We are given two triangles, $\triangle RSV$ and $\triangle RTU$, and we need to explain why these two triangles are similar, and then find the length of segment $RT$.

Similarity of $\triangle RSV$ and $\triangle RTU$:

The similarity between two triangles can be established using the Angle-Angle (AA) similarity criterion, which states that two triangles are similar if two pairs of corresponding angles are congruent.

1. Vertical Angles: The angle $\angle S$ is common to both triangles $\triangle RSV$ and $\triangle RTU$. Therefore, $\angle RSV = \angle RTU$.
2. Corresponding Angles: Since $ST$ is a transversal, and $\angle RSV$ and $\angle RTU$ share the same transversal, the angles $\angle RVS$ and $\angle RTU$ are equal due to being vertically opposite.

Thus, the triangles $\triangle RSV$ and $\triangle RTU$ are similar by the AA (Angle-Angle) similarity criterion.

Finding RT:

Since $\triangle RSV \sim \triangle RTU$, the corresponding sides of the triangles are proportional. Let's set up the proportion using the sides we know:

$\frac{RS}{RT} = \frac{SV}{TU}$

From the figure:

• $RS = 10$
• $SV = 8$
• $TU = 12$

Now, using the proportion:

$\frac{10}{RT} = \frac{8}{12}$

Simplifying the right-hand side:

$\frac{10}{RT} = \frac{2}{3}$

Cross-multiplying:

$2 \cdot RT = 3 \cdot 10$ $2 \cdot RT = 30$ $RT = \frac{30}{2} = 15$

Thus, $RT = 15$.

Correct Option:

The correct answer is: $\triangle RSV \sim \triangle RTU$ by AA, $RT = 15$.

Would you like further clarification on this solution?

Here are some related questions:

1. How do you determine if two triangles are congruent versus similar?
2. Can you use the Side-Angle-Side (SAS) similarity criterion in this problem?
3. How does the concept of vertical angles help in proving similarity?
4. What are some other methods to establish triangle similarity?
5. How would the problem change if $TU$ were different?

Tip: Always look for angles that are common between two triangles when attempting to prove similarity!