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In the given diagram, we know that:

- \( m \angle 2 = 52^\circ \).

Since angles 2 and 4 are vertical angles, they are equal. Thus:

\[
m \angle 4 = m \angle 2 = 52^\circ
\]

Angles 1 and 2 form a linear pair (they are supplementary since they are on a straight line). Therefore:

\[
m \angle 1 + m \angle 2 = 180^\circ
\]

Substitute the value of \( m \angle 2 \):

\[
m \angle 1 + 52^\circ = 180^\circ
\]
\[
m \angle 1 = 180^\circ - 52^\circ = 128^\circ
\]

Similarly, angles 1 and 3 are vertical angles, so they are also equal:

\[
m \angle 3 = m \angle 1 = 128^\circ
\]

### Summary
- \( m \angle 1 = 128^\circ \)
- \( m \angle 3 = 128^\circ \)
- \( m \angle 4 = 52^\circ \)

Would you like further details or have any questions about this solution?

### Related Questions
1. How can we determine if two angles are vertical?
2. What are linear pairs of angles and how do they relate to supplementary angles?
3. Why are vertical angles always equal?
4. What is the relationship between complementary and supplementary angles?
5. How would the solution change if \( m \angle 2 = 60^\circ \) instead?

### Tip
Remember that vertical angles are always equal, and angles on a straight line add up to \(180^\circ\). This helps in finding unknown angles in intersecting lines.

In the figure below, m∠2 = 52°. Find m∠1, m∠3, and m∠4.

Learn how to find unknown angles in intersecting lines when given m∠2 = 52°. This problem uses concepts of vertical angles and linear pairs to solve for m∠1, m∠3, and m∠4. Suitable for students in Grades 6-8.

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