Answer
- Prove $\angle DEF\cong\angle CBA$
- Show from given that $\overline{FD}\cong\overline{AC}$
- Prove $\angle DFE\cong\angle CAB$
- Then $\triangle FED\cong\triangle ABC$ according to method AAS
- Then $\overline{FE}\cong\overline{AB}$ according to CPCTC
Work Step by Step
*PLANNING:
We would prove $\triangle FED\cong\triangle ABC$ according to method AAS:
- Prove $\angle DEF\cong\angle CBA$
- Show from given that $\overline{FD}\cong\overline{AC}$
- Prove $\angle DFE\cong\angle CAB$
1. $\overline{DE}\bot\overline{EF}$ and $\overline{CB}\bot\overline{AB}$. (Given)
2. $\angle DEF$ and $\angle CBA$ are right angles (the angle created from perpendicular lines are right angle)
3. $\angle DEF\cong\angle CBA$ (2 corresponding right angles are congruent)
4. $\overline{FD}\cong\overline{AC}$ (Given)
5. $\overline{AB}\parallel\overline{FE}$ (Given)
6. $\angle DFE\cong\angle CAB$ (2 alternate interior angles for 2 parallel lines are congruent)
So now we have 2 angles and a non-included side of $\triangle FED$ are congruent with 2 corresponding angles and a non-included side of $\triangle ABC$. Therefore,
6. $\triangle FED\cong\triangle ABC$ (AAS)
7. $\overline{FE}\cong\overline{AB}$ (CPCTC)
