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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.

If lines m and n are parallel in the illustration below, which of the following must be true about ∠A and ∠B?

Learn how to identify angle relationships when parallel lines are cut by a transversal. This problem explores the properties of same-side interior angles, showing that they are supplementary. Suitable for Grades 8-10 geometry students.

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