To prove that $m \parallel q$ given $m \parallel n$ and $n \parallel q$, we'll use the concept of corresponding angles formed by parallel lines and a transversal.

Steps to Prove $m \parallel q$:

1. Identify Corresponding Angles:

   • Since $m \parallel n$, by the Alternate Interior Angles Theorem, we have: $\angle 1 \cong \angle 3 \quad \text{and} \quad \angle 2 \cong \angle 4$
   • Since $n \parallel q$, we also have: $\angle 5 \cong \angle 7 \quad \text{and} \quad \angle 6 \cong \angle 8$

2. Establish Angle Relationships:

   • Notice that angles $\angle 3$ and $\angle 5$ are both on the same side of transversal $n$, and since $m \parallel n \parallel q$, these angles are congruent because they correspond to the same angle on parallel lines $m$ and $q$.
   • Therefore: $\angle 3 \cong \angle 5$

3. Conclusion:

   • Since $\angle 3 \cong \angle 5$ and these are corresponding angles formed by the transversal $n$ with lines $m$ and $q$, it follows that $m \parallel q$.

Thus, by the Corresponding Angles Postulate, $m \parallel q$ is proven.

Would you like more details on any specific part of this proof? Here are some related questions:

1. What are alternate interior angles and how are they used in proving parallel lines?
2. Can you explain how corresponding angles help in proving lines are parallel?
3. How does the Transitive Property of parallel lines work in geometry?
4. What is the significance of the Corresponding Angles Postulate?
5. Could you provide another example of proving lines are parallel using different angles?

Tip: Always remember that corresponding angles are crucial in identifying parallel lines, especially when working with multiple transversals.