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To solve this problem, let's analyze the relationships between the angles.

In the diagram, we have two intersecting lines forming vertical angles. Here’s a breakdown of the relationships between the angles:

1. **Given:** \( m \angle 4 = 41^\circ \).
2. **Vertical Angles:** Angles that are opposite each other when two lines intersect are called vertical angles, and they are always equal.

Since \( m \angle 4 = 41^\circ \), it follows that:
\[
m \angle 2 = 41^\circ
\]

3. **Linear Pair:** Angles that are adjacent and form a straight line add up to \(180^\circ\). Thus:
   - \( m \angle 1 + m \angle 2 = 180^\circ \)
   - \( m \angle 3 + m \angle 4 = 180^\circ \)

Since \( m \angle 2 = 41^\circ \), we can find \( m \angle 1 \) as follows:
\[
m \angle 1 = 180^\circ - 41^\circ = 139^\circ
\]

Similarly, since \( m \angle 4 = 41^\circ \), we can find \( m \angle 3 \) as follows:
\[
m \angle 3 = 180^\circ - 41^\circ = 139^\circ
\]

### Summary of Solutions:
- \( m \angle 1 = 139^\circ \)
- \( m \angle 2 = 41^\circ \)
- \( m \angle 3 = 139^\circ \)

Would you like more details or have any questions?

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Here are some related questions to deepen understanding:

1. What are supplementary angles, and how are they used in this problem?
2. How would the solution change if \( m \angle 4 \) were a different value?
3. Why do vertical angles have the same measurement?
4. How can we verify that two lines are intersecting perpendicular to each other based on angle measures?
5. How do linear pairs help in solving for unknown angles?

**Tip:** Always check for vertical angles and linear pairs when solving for angles in intersecting lines. They provide quick solutions.

In the figure below, m∠4 = 41°. Find m∠1, m∠2, and m∠3.

Learn how to find angles m∠1, m∠2, and m∠3 in a geometric figure with intersecting lines using the Vertical Angles Theorem and Linear Pair Postulate. Given m∠4 = 41°, this solution explains each step to determine the remaining angles.

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