Parallelograms & Triangles between Parallels

In the diagram above, the four shapes lie between the same parallels \( \scriptsize \bar{UV} \) and \( \scriptsize \bar{XY} \) i.e. \( \scriptsize \bar{UV} || \bar{XY} \), the parallelograms ABCD and EFGH have their bases on \( \scriptsize \bar{XY} \) and their opposite sides on \( \scriptsize \bar{UV} \), also the triangles JKL and MNO are drawn with their bases on \( \scriptsize\bar{XY} \)and their opposite vertices on \(\: \scriptsize \bar{UV} \).

Note: that the altitudes of the four shapes are all equal (h). The altitude is the distance between the two parallel lines.

The following theorems have been proven to be true for parallelograms and triangles.

i. Parallelograms on the same base and between the same parallels are equal in area.

i.e. Area of ABCD = Area of PQCD.