The problem provided is a geometry question involving parallel lines and angles formed by a transversal. Let's analyze the situation.

Given:

• Lines m and n are parallel.
• The transversal crosses these lines and forms angles ∠A and ∠B at the points shown.

Now, we need to determine the correct relationship between ∠A and ∠B from the following options:

• A. $m∠A = m∠B$
• B. $m∠A = \frac{1}{2}(m∠B)$
• C. $m∠A + m∠B = 90^\circ$
• D. $m∠A + m∠B = 180^\circ$

Key Concept:

When two parallel lines are cut by a transversal, several types of angle relationships occur:

1. Corresponding angles are congruent.
2. Alternate interior angles are congruent.
3. Same-side (consecutive) interior angles are supplementary (i.e., they add up to $180^\circ$).

Here, ∠A and ∠B are on the same side of the transversal and between the two parallel lines, which makes them consecutive interior angles. Therefore, they must be supplementary.

Conclusion:

• The correct option is D: $m∠A + m∠B = 180^\circ$.

Let me know if you'd like further details or have any questions!

Related questions:

1. What is the relationship between alternate interior angles when a transversal crosses parallel lines?
2. How do corresponding angles behave with parallel lines?
3. How can you prove that same-side interior angles are supplementary?
4. What happens to the angles if the lines are not parallel?
5. Can angles formed by non-parallel lines and a transversal still be congruent in some cases?

Tip:

Always remember that when two parallel lines are cut by a transversal, alternate interior and corresponding angles are congruent, while consecutive interior angles are supplementary.