10. (10 points) Taking 2 angles from the 4 interior angles of a quadrilateral to find their sum, there are 6 sum values in total, then the number of sums greater than $180^{\circ}$ is at most $\qquad$.


【Analysis】Let the four angles be $A, B, C, D$, then $A+B+C+D=360^{\circ}$, 6 sums are: $A+C, A+B, A+D$, $B+C, B+D, C+D$, and can be divided into three groups for discussion.
【Solution】Solution: Let the four angles be $A, B, C, D$,
then $A+B+C+D=360^{\circ}$,
6 sums are: $A+C, A+B, A+D, B+C, B+D, C+D$,
divided into three groups:
$$
\begin{array}{l}
A+B \rightarrow C+D, A+B>180^{\circ} \Rightarrow C+D<180^{\circ}, \\
A+D \rightarrow B+C, A+D>180^{\circ} \Rightarrow B+C<180^{\circ}, \\
A+C \rightarrow B+D, A+C>180^{\circ} \Rightarrow B+D<180^{\circ},
\end{array}
$$

Therefore, the number of sums greater than $180^{\circ}$ is at most 3.
The answer is: 3.