10. (10 points) There is a piece of paper in the shape of an isosceles trapezoid, with the upper base length being 2015 and the lower base length being 2016. Using this piece of paper, cut out some isosceles trapezoids, requiring that the bases of the cut trapezoids lie on the bases of the original trapezoid, and the acute angles of the cut trapezoids are equal to the acute angles of the original trapezoid. What is the maximum number of identical isosceles trapezoids that can be cut out?
4029
【Analysis】Given an isosceles trapezoidal piece of paper, the upper base length is 2015, and the lower base length is 2016, so the difference between the upper and lower bases is 1. We need to cut out trapezoids such that the two bases of the cut trapezoids lie on the original trapezoid's bases, and the two acute angles of the cut trapezoids are equal to the acute angles of the original trapezoid. Therefore, the length of the lower base of the cut trapezoid is approximately greater than \(2016 - 2015 = 1\). Based on this, we can solve the problem.

【Solution】Solution: \((2015 - 1) \times 2 + 1\)
$$
\begin{array}{l}
=2014 \times 2 + 1 \\
=4028 + 1 \\
=4029 \text { (pieces) }
\end{array}
$$

Answer: The maximum number of identical isosceles trapezoids that can be cut out is 4029.
Therefore, the answer is: 4029.