In this geometry problem, lines $l_1$ and $l_2$ are parallel, and they are intersected by transversals $p$ and $q$. The measures of angles 2 and 13 are provided as 42° and 58°, respectively, and we are asked to find the measure of angle 8.

Let's break it down step by step:

1. Parallel Line Properties: Since $l_1$ and $l_2$ are parallel, corresponding, alternate interior, and vertically opposite angles are equal.

2. Angle 2 (42°): Angle 2 corresponds to angle 6 (since they are on the same side of the transversal and in corresponding positions). Therefore: $\text{Angle 6} = 42°$

3. Angle 13 (58°): Angle 13 is vertically opposite angle 12, so: $\text{Angle 12} = 58°$

4. Using Linear Pair Rule: Angles 6 and 8 are supplementary (since they form a linear pair along line $q$), meaning: $\text{Angle 6} + \text{Angle 8} = 180°$ Substituting the value of angle 6: $42° + \text{Angle 8} = 180°$ Solving for angle 8: $\text{Angle 8} = 180° - 42° = 138°$

Thus, the measure of angle 8 is 138°.

Do you want further clarification or more details? Here are a few related questions to expand on the topic:

1. What is the relationship between angles 1 and 5 in the figure?
2. How do alternate interior angles work in this setup?
3. How can we find the measure of angle 7 using the same principles?
4. What would change if lines $l_1$ and $l_2$ were not parallel?
5. How do you identify corresponding angles in any parallel line diagram?

Tip: Always check whether lines are parallel to use angle properties like alternate interior angles or corresponding angles effectively.