Theorem 14.1.4

Theorem: Triangles on the equal bases and of equal altitudes are equal in area.

Given: Triangle ABC and DEF are on the equal bases $\overline{BC}$ and $\overline{EF}$ respectively and having equal altitude.

To Prove:

$\blacktriangle ABC = \blacktriangle DEF$.

Construction:

Draw $\overleftrightarrow{AD}$, $\overleftrightarrow{BF}$ containing points B, C, E, F. Place the triangle ABC and DEF so that their equal bases $\overline{BC}$ and $\overline{EF}$ are on the straight line. Draw $\overline{BX}||\overline{CA}$ and $\overline{FY}||\overline{ED}$. Such that point X and Y lie on $\overline{AD}$.

Proof: