5. In $\triangle A B C$, $A D, B E$ are the altitudes on sides $B C, A C$, with $D, E$ being the feet of the perpendiculars. If $C E+C D=A B$, then $\angle C$ is ( ).
(A) acute angle
(B) right angle
(C) obtuse angle
(D) acute angle, right angle, obtuse angle are all possible
5. (A).

If $\angle C=90^{\circ}$, then points $C, D, E$ coincide, which contradicts $C E+C D = A B$.

If $\angle C$ is an obtuse angle, then $D$ is on the extension of $B C$, and $E$ is on the extension of $A C$. By the property that in a right triangle the hypotenuse is the longest side, we have
$$
\begin{array}{l}
\quad A B>B D=B C \\
+C D>C E+C D
\end{array}
$$

which contradicts the given condition. Eliminating (B), (C), and (I), we conclude that (A) is true.