The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?
1. **Identify the vertices and diagonals:**
   - The vertices of the regular decagon are labeled \(D_1, D_2, D_3, \ldots, D_{10}\).
   - We need to find the angle between the diagonals \(D_1D_3\) and \(D_2D_5\).

2. **Understand the properties of a regular decagon:**
   - A regular decagon is a cyclic polygon, meaning all its vertices lie on a common circle.
   - The central angle subtended by each side of the decagon is \(\frac{360^\circ}{10} = 36^\circ\).

3. **Calculate the central angles subtended by the diagonals:**
   - The diagonal \(D_1D_3\) subtends an arc of 2 sides, so the central angle \( \angle D_1OD_3 \) is \(2 \times 36^\circ = 72^\circ\).
   - The diagonal \(D_2D_5\) subtends an arc of 3 sides, so the central angle \( \angle D_2OD_5 \) is \(3 \times 36^\circ = 108^\circ\).

4. **Determine the angle between the diagonals:**
   - The angle between the diagonals \(D_1D_3\) and \(D_2D_5\) is half the sum of the central angles they subtend.
   - Therefore, the angle between the diagonals is:
     \[
     \frac{1}{2} (\angle D_1OD_3 + \angle D_2OD_5) = \frac{1}{2} (72^\circ + 108^\circ) = \frac{1}{2} \times 180^\circ = 90^\circ
     \]

The final answer is \(\boxed{90^\circ}\)