Here are the answers to each of the questions based on the diagram:

1. Transversal Line:

   • Answer: Line $a$ is the transversal.
   • Explanation: A transversal is a line that crosses two or more other lines. In this case, line $a$ crosses lines $m$ and $n$, making it the transversal.

2. Vertical Angles:

   • Answer: Angles $3$ and $7$ (or $4$ and $8$).
   • Explanation: Vertical angles are pairs of opposite angles made by two intersecting lines. Angles $3$ and $7$ (as well as $4$ and $8$) are congruent because vertical angles are always congruent.

3. Alternate Interior Angles:

   • Answer: Angles $3$ and $5$ (or $4$ and $6$).
   • Explanation: Alternate interior angles are on opposite sides of the transversal and lie between the two lines. Angles $3$ and $5$ are congruent, as are $4$ and $6$, according to the alternate interior angle theorem.

4. Same-Side Exterior Angles:

   • Answer: Angles $2$ and $8$ (or $1$ and $7$).
   • Explanation: Same-side exterior angles lie on the same side of the transversal and outside the parallel lines. These angles are supplementary, meaning they add up to $180^\circ$, according to the same-side exterior angle theorem.

5. Corresponding Angles:

   • Answer: Angles $1$ and $5$ (or $2$ and $6$, $3$ and $7$, $4$ and $8$).
   • Explanation: Corresponding angles are on the same side of the transversal and in matching positions relative to the two lines. These angles are congruent because corresponding angles are always congruent when the lines are parallel.

Would you like additional details on any of these explanations or have any questions?

Here are some related questions for further practice:

1. What happens to the angle relationships if lines $m$ and $n$ are not parallel?
2. How do we determine if lines $m$ and $n$ are parallel just by looking at the angles created with the transversal?
3. Can you identify another pair of vertical angles in the diagram?
4. What are alternate exterior angles in this diagram, and are they congruent or supplementary?
5. How can you prove that same-side interior angles are supplementary using the properties of a transversal?

Tip: Remember, vertical angles are always congruent, and corresponding angles are congruent when the lines are parallel.