Since $\angle PQR = \angle PRQ$ and $\angle STR = \angle TSR$, we observe that both $\triangle PQR$ and $\triangle STR$ are isosceles.

Step 1: Analyzing $\triangle PQR$

Given that $\triangle PQR$ is isosceles with $\angle PQR = \angle PRQ$, we have $PQ = PR$. Therefore, if we find $PQ$, we also know $PR$.

Step 2: Analyzing $\triangle STR$

Since $\angle STR = \angle TSR$, $\triangle STR$ is also isosceles with $ST = SR$. Given that $RQ = 8$ and $SQ = 3$, we can deduce that $SR = RQ - SQ = 8 - 3 = 5$. Thus, $SR = 5$ and $ST = 5$.

Step 3: Identifying Similarity between $\triangle PQR$ and $\triangle STR$

Since $\angle PQR = \angle STR$ and $\angle PRQ = \angle TSR$, we can conclude that $\triangle PQR \sim \triangle STR$ by the AA (Angle-Angle) similarity criterion.

Step 4: Setting Up the Proportion

Because $\triangle PQR \sim \triangle STR$, the corresponding sides are proportional: $\frac{PQ}{ST} = \frac{QR}{TR}$ Substituting the known values: $\frac{PQ}{5} = \frac{8}{5}$

Step 5: Solving for $PQ$

From the proportion, we find: $PQ = \frac{8}{5} \times 5 = 8$

Conclusion

Thus, the length of $PQ$ is: $\boxed{8}$

Would you like further explanation on any part? Here are some questions for further exploration:

1. How do we determine similarity between triangles?
2. What properties do isosceles triangles have?
3. How do we set up proportions in similar triangles?
4. How does angle equality help identify similar triangles?
5. Why is it important to identify isosceles triangles in this problem?

Tip: In geometry problems involving isosceles triangles, look for symmetry and congruent angles, as they often reveal proportional relationships.