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The problem involves a rectangular piece of paper \(ABCD\) folded along the diagonal \(BD\), and you are asked to find the angle \(\angle EBC\).

### Analysis of the diagram:

- The rectangle \(ABCD\) is folded along the diagonal \(BD\), and the paper is folded such that point \(E\) (a new point created by the fold) is introduced.
- The angle \(\angle DAE = 55^\circ\) is marked.
- You are tasked with finding \(\angle EBC\).

### Steps to solve:

1. **Rectangular properties**:
   - Since \(ABCD\) is a rectangle, we know that all internal angles are \(90^\circ\).
   - The diagonals of a rectangle are congruent, so \(BD\) bisects \(\angle ABC\), making \(\angle ABD = \angle DBC = 45^\circ\).

2. **Folding property**:
   - Folding the paper along diagonal \(BD\) means that triangle \(ADE\) is reflected over diagonal \(BD\). As a result, \(\angle EBD = \angle ABD = 45^\circ\).

3. **Angle relationships**:
   - To find \(\angle EBC\), we need to analyze the angles around point \(B\).
   - We know that \(\angle ABC = 90^\circ\) (since it’s a corner of the rectangle).
   - From the reflection and folding, \(\angle EBD = 45^\circ\), and \(\angle DBC = 45^\circ\).

4. **Final calculation**:
   \[
   \angle EBC = \angle ABC - \angle EBD = 90^\circ - 45^\circ = 45^\circ.
   \]

Thus, the angle \(\boxed{45^\circ}\) is the measure of \(\angle EBC\).

Let me know if you'd like further details or explanations!

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Here are 5 related questions for deeper understanding:

1. How do we calculate angles in folded paper problems involving symmetry?
2. Why does the diagonal of a rectangle bisect the angles of the rectangle?
3. How does folding along a diagonal affect angle reflections?
4. What is the relationship between folding and congruent triangles?
5. Can we generalize the approach used here to other shapes?

**Tip:** Always look for symmetry in problems involving folding—this often helps simplify the angle relationships!

A rectangular piece of paper ABCD is folded along BD. Find ∠EBC.

Solve the problem of finding ∠EBC in a folded rectangular paper ABCD along diagonal BD. This involves understanding the properties of rectangles, angle bisectors, and reflection symmetry. Suitable for students in Grades 9-11.

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