Parallelograms & Triangles between Parallels

Introduction:

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

The following theorems have been proven to be true for parallelograms and triangles between the same parallel lines:

(a) Equality of the Areas of Two Parallelograms:

Parallelograms between a pair of parallel lines have the same area when their bases are the same length or when they share a common base.

We see that parallelograms ABDC and PQDC are between the same pair of parallel lines, so their heights are both equal to h.