Guided Example #2 - Parallelism and Euclidean Geometry

In the diagram, B, D, C, and E are collinear points.

Given:

           $\bar{A B}$ is parallel to $\bar{D F}$ ,
            $\bar{A C}$ is parallel to $\bar{E F}$ ,
            BD = CE.

Prove
AB = FD

Statement	Reason
BC = BD + DC and DE = DC + CE

Statement	Reason
BC = BD + DC and DE = DC + CE
Since $B D = C E, \bar{B C} ≅ \bar{D E}$
So $Δ A B C ≅ Δ F D E$	(Angle - Side - Angle Theorem)
Then, $A B = F D$	They are corresponding parts of congruent $Δ$ .