14. B. Color the five sides and five diagonals of the convex pentagon $A B C D E$, such that any two segments sharing a common vertex are of different colors. Find the minimum number of colors needed.
14. B. Since vertex $A$ is the common point of four segments $A B$, $A C$, $A D$, and $A E$, at least four colors are needed.

If only four colors are used, let's assume they are red, yellow, blue, and green. Then, the four segments extending from each vertex must include one each of red, yellow, blue, and green. Therefore, the number of red segments would be $\frac{5}{2}$, which is a contradiction. Thus, at least five colors are needed.

The following example shows that five colors can be used to color the ten segments to meet the conditions: color $A B$ and $C E$ with color 1; color $B C$ and $D A$ with color 2; color $C D$ and $E B$ with color 3; color $D E$ and $A C$ with color 4; color $E A$ and $B D$ with color 5. In this way, any two segments sharing a common vertex will have different colors.
In conclusion, the minimum number of colors required is 5.
(Provided by Ben Guo Min)