We claim that any even sided polygon with all equal lengths and parallel opposite sides (we call it a nice polygon) can be dissected into lozenges (rhombus). Clearly quadrilaterals works, now suppose it works until it has $2n-2$ sides with $n\geq 3$. Now consider a nice $(2n)$-gon $A_1...A_{2n}$. Now let $B_2,B_3,...,B_{n-1}$ inside the polygon such that the line segments $A_2B_2,...,A_{n-1}B_{n-1}$ are equal in length and parallel to $A_{2n}A_1$. Then the quadrilaterals $A_{2n}A_1A_2B_2,B_2A_2A_3B_3,B_3A_3A_4B_4,...,B_{n-1}A_{n-1}A_nA_{n+1}$ are all rhombus. Dissect them away and the remaining figure is a nice $(2n-2)$-gon. Thus by induction all nice polygon can be dissected into rhombus.