jmc

From each vertex $V$, we can draw 2 diagonals: one to each vertex that is not $V$ and does not share an edge with $V$. There are 5 vertices in a pentagon, so we might be tempted to say the answer is $5\times 2=10$. However, note that this counts each diagonal twice, one time for each vertex. Hence there are $\frac{10}{2}=\boxed{5}$ distinct diagonals in a convex pentagon.