5. A diagonal of a convex polygon (i.e., a polygon where all interior angles are less than $180^{\circ}$) is called "bisecting" if and only if the diagonal simultaneously bisects the area and the perimeter of the polygon. How many diagonals of a convex pentagon can be bisecting at most?

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5. In a convex pentagon, at most two diagonals are bisecting.

Notice that, for any vertex of a convex polygon, there is at most one diagonal passing through that vertex that is bisecting. Therefore, in a convex pentagon, the number of bisecting diagonals does not exceed two. For example, in Figure 11, the diagonals \(AD\) and \(BC\) of the convex pentagon are bisecting.