### Instruction: A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees.  A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices.  How many diagonals does a convex polygon with 20 sides have?### Response: I want to find a general formula for the number of diagonals of a convex polygon with n sides.
I notice that every vertex of the polygon can be connected to n - 3 other vertices by a diagonal, since I have to avoid the two adjacent vertices and the vertex itself.
However, this would count each diagonal twice, once for each endpoint.
So I need to divide by 2 to get the actual number of diagonals.
Therefore, the formula is n(n - 3) / 2.
Now I can plug in n = 20 and simplify.
20(20 - 3) / 2 = 20(17) / 2 = 10(17) = 170.