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To solve this problem, we need to find the angle \( B \) such that all the sides labeled \( x \) are equal. Let's analyze the diagram step-by-step:

1. **Understanding the diagram**: The diagram shows a rectangle with sides labeled \( x \) and a diagonal line segment inside the rectangle. This diagonal forms a right triangle with one side labeled as 15 and the other side along the rectangle’s length \( x \).

2. **Right Triangle Properties**: The diagonal inside the rectangle can be seen as the hypotenuse of a right triangle where one leg is \( x \) and the other is also \( x \), making this an isosceles right triangle.

3. **Using the Pythagorean Theorem**: In an isosceles right triangle with legs of length \( x \), the hypotenuse (which is the diagonal inside the rectangle) would be \( \sqrt{x^2 + x^2} = x\sqrt{2} \).

4. **Matching the Hypotenuse**: Given that the hypotenuse (the diagonal) is labeled as 15, we equate this to \( x\sqrt{2} \):
   \[
   x\sqrt{2} = 15
   \]
   Solving for \( x \):
   \[
   x = \frac{15}{\sqrt{2}} = \frac{15\sqrt{2}}{2}
   \]

5. **Finding the Angle \( B \)**: Angle \( B \) can be found using the trigonometric relationship within the triangle. Since it’s a 45-45-90 triangle:
   \[
   \tan(B) = \frac{x}{x} = 1
   \]
   \[
   B = \tan^{-1}(1) = 45^\circ
   \]

Thus, for \( x \) to be the same for all sides and the hypotenuse to be 15, angle \( B \) must be \( 45^\circ \).

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What would happen if the hypotenuse was a different length, say 20?
2. How do we calculate the diagonal of a rectangle using its side lengths?
3. How can the Pythagorean theorem be used in other geometric shapes?
4. How would you find \( x \) if angle \( B \) was not \( 45^\circ \)?
5. How do different trigonometric functions relate to right triangles?

**Tip:** Always check the geometric properties and relationships between angles and sides when solving for unknowns in diagrams.

Learn how to find angle B in a rectangle where all sides labeled x are equal. This geometry problem involves using the Pythagorean theorem and understanding the properties of 45-45-90 triangles. Suitable for students in Grades 9-12.

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