Answer
1. We are told that AB is congruent to DE.
2. We are also told that AC is congruent to DF.
3. Since AB is parallel to DE, angle A is congruent to angle D, for if two alternate interior angles formed by parallel lines have the same transversal, they are congruent.
4. By SAS, this means that the two triangles are congruent.
5.Thus, FE and BC are at the same slope, for when two triangles are congruent, it follows that corresponding sides B and E are also congruent.
6. Lines at the same slope are parallel, so FE and BC are parallel.
Work Step by Step
Using the direct proof:
1.Given $\overline{AB}=\overline{DE} $
$ \overline{AB\parallel} \overline{DE}$
$ \overline{AC}=\overline{DF}$
2. $ \angle BAC = \angle EDF $ if two parallel lines are cut by a transversal then the two alternate interior angles are congruent.
3. $ \triangle ABC= \triangle DEF $ by SAS
4. $ \angle BCA = \angle EFD $ by CPCTC
5. $\overline{BC} \parallel \overline{FE}$
If two lines are cut by a transversal so that two alternate interior angles are congruent then these lines are parallel. $ \square$
