To complete this flowchart proof, let’s analyze the given information and the structure provided:

Given:

1. $AB \parallel DC$
2. $BC \parallel AD$

Flowchart Structure:

1. First Step:

   • Statements: $AB \parallel DC$ and $BC \parallel AD$
   • Reasons: Both are given in the problem statement.

2. Second Step:

   • Statement: $\angle BAC \cong \angle DCA$
   • Reason: Since $AB \parallel DC$, and $AC$ is a transversal, these angles are Alternate Interior Angles and therefore congruent.

3. Third Step:

   • Statement: $\angle BCA \cong \angle DAC$
   • Reason: Since $BC \parallel AD$, and $AC$ is a transversal, these angles are also Alternate Interior Angles and therefore congruent.

4. Fourth Step:

   • Statement: $BC \cong AD$
   • Reason: By the Given information in the problem statement.

5. Final Step:

   • Statement: $\triangle ABC \cong \triangle CDA$
   • Reason: The triangles are congruent by the ASA (Angle-Side-Angle) Congruence Postulate, as we have two pairs of congruent angles and the included side $BC \cong AD$.

Would you like a more detailed breakdown of each reasoning step, or do you have further questions about this proof?

Follow-up Questions:

1. Why are alternate interior angles congruent when lines are parallel?
2. What does the ASA (Angle-Side-Angle) Congruence Postulate state in general?
3. Can we prove congruence with a different postulate here, or is ASA the only option?
4. How can we apply similar logic to prove congruence in other parallelogram shapes?
5. If a shape has one pair of opposite angles congruent, does that imply it’s a parallelogram?

Tip:

When dealing with parallel lines and a transversal, always look for alternate interior angles, corresponding angles, or consecutive interior angles, as they often lead to key congruences in proofs.