To solve this problem, we use the concept of similar triangles, as the angles $\angle PQO \cong \angle MNO$ indicate that triangles $\triangle PQO$ and $\triangle MNO$ are similar.

Steps to Solve:

1. Set up the similarity ratio: Since $\triangle PQO \sim \triangle MNO$, the corresponding sides of the triangles are proportional. This gives: $\frac{PQ}{MN} = \frac{OQ}{ON}$

2. Substitute known values: We are given:

   • $PQ = 6$,
   • $OQ = 12$,
   • $ON = 20$.

   Substituting these into the ratio: $\frac{6}{MN} = \frac{12}{20}$

3. Simplify the ratio: Simplify $\frac{12}{20}$ to $\frac{3}{5}$. The equation becomes: $\frac{6}{MN} = \frac{3}{5}$

4. Solve for $MN$: Cross-multiply to find $MN$: $6 \cdot 5 = 3 \cdot MN$ $30 = 3 \cdot MN$ $MN = \frac{30}{3} = 10$

Final Answer:

The length of $MN$ is: $\boxed{10}$

Would you like a detailed explanation of any step? Let me know!

Here are 5 related questions to explore:

1. What are the criteria for two triangles to be similar?
2. How do angle congruences affect triangle similarity?
3. If the length of $ON$ changes, how does it affect $MN$?
4. Can the proportionality method be used for non-right triangles?
5. How does the concept of similarity apply in real-life problems?

Tip: Always simplify fractions in similarity problems to make proportional calculations easier.