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!

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!