Let's break down the steps to complete the proof of the given theorem:

Problem:

We need to prove that quadrilateral $ABCD$ is a parallelogram given that:

• $\overline{AB} \cong \overline{CD}$
• $\overline{AD} \cong \overline{BC}$

By definition, a parallelogram is a quadrilateral where both pairs of opposite sides are parallel. Using this definition, let's proceed step-by-step.

Proof:

1. Statement: $\overline{AB} \cong \overline{CD}$

   • Reason: Given

2. Statement: $\overline{AD} \cong \overline{BC}$

   • Reason: Given

3. Statement: $\angle ABD \cong \angle DCB$

   • Reason: Opposite angles of a parallelogram are congruent.

4. Statement: $\angle 1 \cong \angle 2$

   • Reason: Vertical angles are congruent (since $\angle ABD$ and $\angle DCB$ are vertical angles).

5. Statement: $\angle 1 \cong \angle 3$

   • Reason: Alternate interior angles are congruent when parallel lines are cut by a transversal.

6. Statement: $\angle 2 \cong \angle 4$

   • Reason: Alternate interior angles are congruent when parallel lines are cut by a transversal.

7. Statement: $\overline{AB} \parallel \overline{CD}$

   • Reason: If alternate interior angles $\angle 1$ and $\angle 3$ are congruent, then lines $\overline{AB}$ and $\overline{CD}$ are parallel.

8. Statement: $\overline{AD} \parallel \overline{BC}$

   • Reason: If alternate interior angles $\angle 2$ and $\angle 4$ are congruent, then lines $\overline{AD}$ and $\overline{BC}$ are parallel.

9. Statement: $ABCD$ is a parallelogram

   • Reason: By definition, a quadrilateral with both pairs of opposite sides parallel is a parallelogram.

If you need further explanation or have any additional questions, feel free to ask!

Here are some related questions for your understanding:

1. How do alternate interior angles help prove lines are parallel?
2. What other properties can be used to prove a quadrilateral is a parallelogram?
3. Can a quadrilateral be a parallelogram if only one pair of opposite sides is parallel?
4. How would the proof change if the given were different angles or side lengths?
5. What are some practical examples of parallelograms in real life?

Tip: When working with parallelograms, remember that opposite angles are congruent, and opposite sides are both congruent and parallel. This can simplify proofs significantly!