To prove that triangles $\triangle ABE$ and $\triangle CDE$ are congruent, let's break down the information provided and apply a congruence theorem.

Given:

• Triangles $\triangle ABE$ and $\triangle CDE$ are shown with:
  • Two pairs of corresponding sides marked as congruent: $AB \cong CD$ and $AE \cong CE$.
  • Vertical angles $\angle AEB$ and $\angle CED$ are formed by intersecting lines, making them congruent (Vertical Angle Theorem).

Proof:

1. Identify Given Congruent Sides:

   • $AB \cong CD$ (given by markings).
   • $AE \cong CE$ (given by markings).

2. Identify Congruent Angles:

   • $\angle AEB \cong \angle CED$ (Vertical Angle Theorem, as they are opposite angles formed by intersecting lines).

3. Apply the SAS (Side-Angle-Side) Congruence Postulate:

   • We have two pairs of congruent sides ($AB \cong CD$ and $AE \cong CE$) and the included angle ($\angle AEB \cong \angle CED$) between those sides.

   According to the Side-Angle-Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Conclusion:

By the SAS Congruence Postulate, $\triangle ABE \cong \triangle CDE$.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What other congruence postulates (besides SAS) could prove triangle congruence?
2. How does the Vertical Angle Theorem assist in proving triangle congruence?
3. What would change if we didn’t know the two sides were congruent?
4. Why is it necessary for the congruent angles to be included between the pairs of congruent sides for SAS?
5. Can congruent triangles always be proven with just side lengths?

Tip: Always look for shared angles or sides when proving triangle congruence, as they can be key elements in your proof.