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In this problem, you are asked to find the measures of angles \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \) given that \( m \angle 4 = 72^\circ \). The diagram shows four angles formed by two intersecting lines, and we can use the relationships between vertical and adjacent angles to solve this.

### Step-by-step Solution:

1. **Vertical angles**: 
   - Vertical angles are congruent. Since \( \angle 4 \) and \( \angle 2 \) are vertical angles, their measures are equal.
     \[
     m \angle 2 = m \angle 4 = 72^\circ
     \]

2. **Linear pair (adjacent angles on a straight line)**:
   - The sum of two adjacent angles forming a straight line is \( 180^\circ \). \( \angle 4 \) and \( \angle 3 \) form a linear pair, so:
     \[
     m \angle 3 + m \angle 4 = 180^\circ
     \]
     \[
     m \angle 3 + 72^\circ = 180^\circ
     \]
     Solving for \( m \angle 3 \):
     \[
     m \angle 3 = 180^\circ - 72^\circ = 108^\circ
     \]

3. **Vertical angles again**:
   - \( \angle 1 \) and \( \angle 3 \) are vertical angles, so they are congruent:
     \[
     m \angle 1 = m \angle 3 = 108^\circ
     \]

### Final answers:
- \( m \angle 1 = 108^\circ \)
- \( m \angle 2 = 72^\circ \)
- \( m \angle 3 = 108^\circ \)

Let me know if you want more details or have questions. Here are some related questions:

1. What are supplementary angles, and how do they differ from complementary angles?
2. How do you identify vertical angles in more complex diagrams?
3. Can the angles formed by two intersecting lines ever all be equal?
4. What is the significance of congruent angles in geometry?
5. How do linear pairs apply to parallel line theorems?

**Tip**: Always remember that vertical angles are congruent, and the sum of angles on a straight line is \( 180^\circ \). This helps in quickly solving problems involving intersecting lines.

In the figure below, m∠4 = 72°. Find m∠1, m∠2, and m∠3.

Solve the geometry problem where two intersecting lines form vertical and adjacent angles. Given m∠4 = 72°, find m∠1, m∠2, and m∠3 using the Vertical Angle Theorem and the Linear Pair Postulate. Suitable for Grades 7-9.

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