Answer
- Prove that $\triangle DAB\cong\triangle CAE$ by method ASA to deduce $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$
- Then show that $\overline{DE}\cong\overline{CB}$
- Then we can prove that $\triangle DEC\cong\triangle CBD$ by method SAS.
Work Step by Step
- Prove that $\triangle DAB\cong\triangle CAE$ by method ASA to deduce $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$
- Then show that $\overline{DE}\cong\overline{CB}$
- Then we can prove that $\triangle DEC\cong\triangle CBD$ by method SAS.
* Prove that $\triangle DAB\cong\triangle CAE$
1) $\overline{DB}\bot\overline{BC}$ and $\overline{CE}\bot\overline{ED}$ (Given)
2) $\angle DBA$ and $\angle CEA$ are right $\angle$s (if 2 lines are perpendicular with each other, then the angles that they make up are right angles)
3) $\angle DBA\cong\angle CEA$ (2 corresponding right angles are congruent)
4) $\overline{AB}\cong\overline{AE}$ (Given)
5) $\angle DAB\cong\angle CAE$ (2 vertical angles are congruent)
So now 2 angles and the included side of $\triangle DAB$ are congruent with 2 corresponding angles and the included side of $\triangle CAE$.
5) $\triangle DAB\cong\triangle CAE$ (ASA)
6) $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$ (CPCTC)
* Prove that $\triangle DCB\cong\triangle CDE$
We see that $\overline{DA}\cong\overline{CA}$ (proved above) and $\overline{AE}\cong\overline{AB}$ (given)
So $\overline{DA}+\overline{AE}\cong\overline{CA}+\overline{AB}$
That means $\overline{DE}\cong\overline{CB}$
7) $\overline{DE}\cong\overline{CB}$ (proved above)
8) $\overline{CE}\cong\overline{DB}$ (proved in 6)
9) $\angle CED\cong\angle DBC$ (proved in 3)
So now 2 sides and the included angle of $\triangle DEC$ are congruent with 2 corresponding sides and the included angle of $\triangle CBD$.
10) $\triangle DEC\cong\triangle CBD$ (SAS)
