In the image, two parallel lines are cut by a transversal, and you are asked to find the measures of angles $\angle 2$ and $\angle 4$, given that the measure of $\angle 5 = 139^\circ$.

Here’s how we can solve it using angle relationships for parallel lines and a transversal:

1. Corresponding Angles:

   • $\angle 5$ and $\angle 1$ are corresponding angles, so $m \angle 1 = 139^\circ$.

2. Alternate Interior Angles:

   • $\angle 1$ and $\angle 2$ are alternate interior angles, which means they are congruent. So, $m \angle 2 = 139^\circ$.

3. Linear Pair:

   • $\angle 4$ and $\angle 5$ form a linear pair. Linear pairs are supplementary, meaning the sum of their measures is $180^\circ$. Therefore, $m \angle 4 = 180^\circ - 139^\circ = 41^\circ$.

Final Answer:

• $m \angle 2 = 139^\circ$
• $m \angle 4 = 41^\circ$

Let me know if you need further clarification or details.

Here are 5 related questions to deepen understanding:

1. How would you identify alternate exterior angles in the same diagram?
2. If $m \angle 6$ was given, how could you find $m \angle 3$?
3. What is the relationship between corresponding angles in parallel lines cut by a transversal?
4. Can you explain how vertical angles relate in this diagram?
5. If $m \angle 3 = 80^\circ$, what would be the measure of $m \angle 7$?

Tip: Remember that corresponding and alternate interior angles are congruent when parallel lines are cut by a transversal. This simplifies finding unknown angles.