The problem asks to find the measures of angles $\angle 2$ and $\angle 3$ given that two parallel lines are cut by a transversal, and $m \angle 8 = 139^\circ$.

Step-by-Step Solution:

1. Identify Angle Relationships:

   • Since the lines are parallel and cut by a transversal, we can apply properties such as alternate interior angles, corresponding angles, and supplementary angles.
   • $\angle 8$ and $\angle 3$ are alternate interior angles. Since alternate interior angles are equal when lines are parallel: $m \angle 3 = m \angle 8 = 139^\circ$

2. Use Supplementary Angle Property:

   • Angles on a straight line are supplementary, meaning they add up to $180^\circ$.
   • $\angle 2$ and $\angle 8$ are supplementary because they form a linear pair: $m \angle 2 + m \angle 8 = 180^\circ$ Substituting $m \angle 8 = 139^\circ$: $m \angle 2 + 139^\circ = 180^\circ$ Solving for $m \angle 2$: $m \angle 2 = 180^\circ - 139^\circ = 41^\circ$

Final Answer:

• $m \angle 2 = 41^\circ$
• $m \angle 3 = 139^\circ$

Do you want any further details or explanations? Here are five questions that extend this topic:

1. What are alternate exterior angles, and how do they relate in parallel lines?
2. Can you describe the relationship between corresponding angles?
3. How do vertical angles behave when parallel lines are cut by a transversal?
4. If the measure of $\angle 1$ is known, how can you find $\angle 7$?
5. What is the significance of supplementary angles in geometric proofs?

Tip: Always identify the angle relationships (alternate, corresponding, or supplementary) when working with parallel lines and a transversal to simplify solving the problem.