Given that $\angle A \cong \angle D$ and $\angle 2 \cong \angle 3$, we want to prove that $\overline{AE} \cong \overline{DE}$. The image also provides a table with some statements and reasons, which we will use to derive the final result.

1. Problem Description

Given that $\angle A \cong \angle D$ and $\angle 2 \cong \angle 3$, we want to prove that $\overline{AE} \cong \overline{DE}$. The image also provides a table with some statements and reasons, which we will use to derive the final result.

2. Solution Steps

Here is a step-by-step explanation of the proof:

1. $\angle A \cong \angle D$ (Given)

2. $\angle 2 \cong \angle 3$ (Given)

3. $\triangle BCE$ is isosceles (Base Angle Theorem: if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Since $\angle 2 \cong \angle 3$, then $\triangle BCE$ is an isosceles triangle.)

4. $\overline{BE} \cong \overline{CE}$ (Definition of Isosceles Triangle: If a triangle is isosceles, then the sides opposite the congruent base angles are congruent.)

5. $\angle 5 \cong \angle 6$ (Vertical Angles Theorem: Vertical angles are congruent.)

6. $\triangle BAE \cong \triangle CDE$ (Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the triangles are congruent. We have $\angle A \cong \angle D$, $\angle 5 \cong \angle 6$, and $\overline{BE} \cong \overline{CE}$. Therefore, $\triangle ABE \cong \triangle DCE$)

7. $\overline{AE} \cong \overline{DE}$ (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

3. Final Answer

$\overline{AE} \cong \overline{DE}$