Let's add the given information to the diagram. At the same time, we will mark two pairs of congruent vertical angles in the middle of the diagram.
We want to prove that △ ABE ≅ △ DEC. The first thing we notice is that two sides in our triangles are congruent, AE≅ EC. Second, by viewing AC as a transversal, we can identify two pairs of alternate interior angles. Since AD∥ BC, we know that these angles are congruent by the Alternate Interior Angles Theorem.
Since two angles and the included side in △ ADE are congruent to two angles and the included side in △ BCE, we can claim that these triangles are congruent by the ASA Congruence Theorem.
Let's mark a second pair of congruent corresponding sides in these triangles that we will need.
Now we have enough information to prove that △ ABE ≅ △ CDE. by the SAS Congruence Theorem.
Proof
Two-Column Proof
Let's show this as a two-column proof as well.
Statement
Reason
1.
&AD∥ BC & Eis the midpoint of AC
1.
Given
2.
AE≅ CE
2.
Definition of midpoint
3.
&∠ AEB ≅ ∠ CED &∠ AED ≅ ∠ CEB
3.
Vertical Angles Congruence Theorem
4.
∠ DAE ≅ ∠ BCE
4.
Alternate Interior Angles Theorem
5.
△ DAE ≅ △ BCE
5.
ASA Congruence Theorem
6.
DE≅ BE
6.
Corresponding parts of congruent triangles are congruent
