https://file.aiquickdraw.com/mathbot/file/17230488575344vk8fomf.jpg

To find the values of \( n \) and \( m \) in the given square, we need to analyze the geometric relationships within the figure. The image shows a square with its diagonals intersecting, creating four right triangles inside the square.

Let's break down the problem:

1. **Square Properties**: 
   - All sides are equal.
   - Each internal angle is \(90^\circ\).
   - The diagonals of a square are equal and bisect each other at right angles (\(90^\circ\)).

2. **Diagonals**:
   - The diagonals of a square split the square into two equal right triangles.

3. **Triangles within the Square**:
   - Each of the right triangles formed by the diagonals has angles \(45^\circ, 45^\circ,\) and \(90^\circ\).

4. **Angle Relationships**:
   - The diagonals intersect at \(90^\circ\) creating angles of \(45^\circ\) at each vertex where they intersect the sides of the square.

Given the symmetry and the right triangle properties:

- **Angles \( n \) and \( m \)**:
   - Angle \( n \) is formed by the intersection of one of the diagonals with a side of the square.
   - Angle \( m \) is another angle formed similarly by the intersection of the diagonals with a side of the square.

### Calculations

From the properties above, let's analyze the angles within the square:

- **Angle \( n \)**:
   - This angle is formed between the diagonal and one side of the square.
   - Since the diagonals bisect each vertex angle of the square (\(90^\circ\)) into two \(45^\circ\) angles, angle \( n \) will be \(45^\circ\).

- **Angle \( m \)**:
   - This angle is also formed between the diagonal and one side of the square.
   - Similarly, angle \( m \) will be \(45^\circ\) for the same reasons outlined above.

Thus, both angles \( n \) and \( m \) are \(45^\circ\) each.

### Conclusion

- \( n = 45^\circ \)
- \( m = 45^\circ \)

Would you like more details on any specific geometric properties or calculations?

1. How do diagonals of a square create symmetry in internal angles?
2. What is the significance of right triangles formed by diagonals in a square?
3. How can bisected angles help determine internal angle values?
4. Why are the angles formed by diagonals in a square always \(45^\circ\)?
5. What are other properties of squares important in angle calculations?
6. How does understanding square properties help in complex geometry problems?
7. What role do supplementary angles play in square geometry?
8. How can geometric proofs validate these angle findings?

**Tip:** Always use the inherent symmetry and properties of geometric shapes to simplify angle and side length calculations.

Explore the geometric properties of squares, focusing on angles and the intersections of diagonals. Learn how to determine internal angle values using symmetry and right triangle properties. Suitable for Grades 7-9 students studying basic geometry.

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