When we scan the answer choices, we see that they are all small values.
So, one option is to simply draw a pentagon and see how many diagonals you can draw.
When we do that we get:

As we can see, there are 5 diagonals.
Answer: A


Of course, the strategy wouldn't work well if the polygon has a lot of size (e.g., a 20-sided polygon).
So here's another approach.
We can see that there are 2 diagonals from vertex A.


In fact all 5 vertices will have 2 diagonals coming from it.
So, the total number of diagonals = (5)(2) = 10
However, we must recognize that we have counted it's diagonal TWICE.
For example, we count the diagonal from A to C as 1 diagonal, and we count the diagonal from C to A as another diagonal.
To account for this duplication, we must divide 10 by 2 to get 5.
Answer: A