To complete this flowchart proof, let's examine each component step-by-step:

Step-by-Step Completion

1. Given:

   • $AB \parallel DC$ (Given)
   • $E$ is the midpoint of $BD$ (Given)

2. Angle Congruence:

   • Since $AB \parallel DC$, $\angle B \cong \angle D$.
   • Reason: Corresponding Angles Postulate (because parallel lines have corresponding angles that are congruent).

3. Segment Congruence (Midpoint):

   • Since $E$ is the midpoint of $BD$, we have $BE \cong ED$.
   • Reason: Definition of Midpoint (a midpoint divides a segment into two congruent parts).

4. Additional Segment Congruence:

   • $AB \cong CD$.
   • Reason: Given.

5. Triangle Congruence:

   • Triangles $\triangle ABE$ and $\triangle CDE$ are congruent.
   • Reason: SAS (Side-Angle-Side) Congruence Postulate. We have two pairs of congruent sides $AB \cong CD$ and $BE \cong ED$, and the included angles $\angle B \cong \angle D$.

Completed Flowchart Proof

This setup should fill in the blanks in the flowchart as follows:

• $\angle B \cong \angle D$
  • Reason: Corresponding Angles Postulate
• $BE \cong ED$
  • Reason: Definition of Midpoint
• $\triangle ABE \cong \triangle CDE$
  • Reason: SAS (Side-Angle-Side) Congruence Postulate

Would you like further details on any part of this proof?

Further Questions

1. Why is the Corresponding Angles Postulate applicable to parallel lines?
2. How does the midpoint definition guarantee segment congruence?
3. What other methods could be used to prove triangle congruence here, if any?
4. Why is SAS specifically chosen over other triangle congruence postulates?
5. Could we have deduced the congruence of $AB$ and $CD$ another way?

Tip

In geometry, parallel lines are often used to establish congruent angles, especially when proving congruence between triangles that share corresponding angles.