For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which
$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
1. **Identify the given points and ratios:**
   - Point \( X \) on side \( AB \) such that \( \frac{AX}{AB} = \frac{4}{5} \).
   - Point \( Y \) on segment \( CX \) such that \( \frac{CY}{YX} = 2 \).
   - Point \( Z \) on the ray \( CA \) such that \( \angle CXZ = 180^\circ - \angle ABC \).

2. **Determine the coordinates of point \( X \):**
   - Let \( A = (0, 0) \) and \( B = (b, 0) \).
   - Since \( \frac{AX}{AB} = \frac{4}{5} \), the coordinates of \( X \) are \( X = \left(\frac{4b}{5}, 0\right) \).

3. **Determine the coordinates of point \( Y \):**
   - Let \( C = (c_x, c_y) \).
   - The line segment \( CX \) can be parameterized as \( C + t(X - C) \) for \( t \in [0, 1] \).
   - Since \( \frac{CY}{YX} = 2 \), we have \( CY = 2YX \), implying \( Y \) divides \( CX \) in the ratio \( 2:1 \).
   - Using the section formula, the coordinates of \( Y \) are:
     \[
     Y = \left( \frac{2 \cdot \frac{4b}{5} + c_x}{3}, \frac{2 \cdot 0 + c_y}{3} \right) = \left( \frac{8b + 5c_x}{15}, \frac{c_y}{3} \right)
     \]

4. **Determine the coordinates of point \( Z \):**
   - Point \( Z \) lies on the ray \( CA \) such that \( \angle CXZ = 180^\circ - \angle ABC \).
   - This implies that \( Z \) is on the extension of \( CA \) past \( A \).

5. **Calculate the angle \( \angle XYZ \):**
   - Given \( \angle XYZ = 45^\circ \), we need to use the properties of the triangle and the given conditions to find the relationship between the angles of the triangle.

6. **Prove similarity of triangles:**
   - To prove that all triangles in \( \Sigma \) are similar, we need to show that the angles of the triangles are the same.
   - Since \( \angle XYZ = 45^\circ \) and the other conditions are met, we can use the properties of similar triangles and angle chasing to show that the triangles are similar.

7. **Find the measure of the smallest angle:**
   - Let \( \angle A = \alpha \), \( \angle B = \beta \), and \( \angle C = \gamma \).
   - Using the given conditions and the fact that the sum of the angles in a triangle is \( 180^\circ \), we can find the measure of the smallest angle.

8. **Conclusion:**
   - By proving the similarity of the triangles and using the given conditions, we can determine the measure of the smallest angle.

The final answer is \( \boxed{15^\circ} \)