How many diagonal lines are in a dodecagon? 

A dodecagon has 12 sides and 12 vertices.  Every combination of two vertices,
except the combinations representing the 12 sides, will represent a diagonal
line.  Therefore

C(12,2) - 12 = 12!/[2!(12-2)!] - 12 = 12!/(2!10!) - 12 = 66 - 12 = 54

How many diagonal lines are in a hexadecagon?

A hexadecagon has 16 sides and 16 vertices.  Every combination of two vertices,
except the conbinations representing the 16 sides, will represent a diagonal
line.  Therefore

C(16,2) - 16 = 16!/[2!(16-2)!] - 16 = 16!/(2!14!) - 16 = 120 - 16 = 104

Edwin