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Let's analyze the given problem.

We are dealing with a triangle XYZ where:
- \(\angle X = 45^\circ\),
- The length of side \(YZ = n\sqrt{2}\),
- The length of side \(XZ = n\sqrt{3}\).

We need to find all possible measures for \(\angle Y\).

### Steps to solve:
1. **Law of Cosines**: Since we know two sides and one angle, we can apply the Law of Cosines to find the unknown angles in the triangle. The formula is:
   \[
   c^2 = a^2 + b^2 - 2ab\cos(C)
   \]
   Here, we know:
   - \(a = YZ = n\sqrt{2}\),
   - \(b = XZ = n\sqrt{3}\),
   - \(\angle X = 45^\circ\), so we can substitute these into the formula.

   For \(\angle Y\), we use the form:
   \[
   \cos(Y) = \frac{b^2 + c^2 - a^2}{2bc}
   \]
   This formula will help us find \(\cos(Y)\).

2. **Law of Sines**: After finding the value of \(\cos(Y)\), we could alternatively use the Law of Sines to verify the results if needed:
   \[
   \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
   \]

Let me calculate the exact value of \(\angle Y\) based on these steps.The calculated measure of \(\angle Y\) is approximately \(71.73^\circ\).

This value doesn't directly match any of the provided answer choices. However, based on the available options, it seems closest to \(60^\circ\) or \(None of these\), since none of the other ranges fit.

Would you like further clarification or a breakdown of any specific part of this solution?

Here are some related questions you might explore:
1. How would the solution change if \(n\) was given explicitly?
2. How can we use the Law of Sines to check consistency?
3. What happens if we solve for \(\angle Z\) using similar methods?
4. Could there be any geometric significance to \(\angle Y\) being close to \(72^\circ\)?
5. Can we apply this process to other triangles with different side lengths?

**Tip**: Always double-check calculations for angle measures when using trigonometric functions to avoid rounding errors.

In triangle XYZ, the measure of angle X is 45°, the length of YZ is n√2, and the length of XZ is n√3. Find all possible measures for angle Y.

Learn how to find angle Y in triangle XYZ using the Law of Cosines, given angle X is 45°, side YZ is n√2, and side XZ is n√3. Detailed explanation of trigonometric principles for Grades 10-12.

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