The image you provided is a geometric proof where we are given a quadrilateral $ABCD$ with diagonal $AC$, and the conditions $AB \cong CD$ and $AD \cong CB$. The task is to prove that quadrilateral $ABCD$ is a parallelogram.

To prove that $ABCD$ is a parallelogram, we can use the following steps:

Proof Outline:

1. Statement 1: $AB \cong CD$ and $AD \cong CB$ (Given)

   • Reason 1: Given.

2. Statement 2: Triangles $\triangle ABD$ and $\triangle CDB$ are congruent by the Side-Side-Side (SSS) Postulate.

   • Reason 2: SSS Postulate (since $AB = CD$, $AD = CB$, and $BD = BD$).

3. Statement 3: $\angle ABD = \angle CDB$ and $\angle ADB = \angle CDB$ (Corresponding parts of congruent triangles are equal, or CPCTC).

   • Reason 3: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

4. Statement 4: $AB \parallel CD$ (By alternate interior angles theorem or the fact that these angles are congruent).

   • Reason 4: Alternate interior angles theorem (since $\angle ABD = \angle CDB$).

5. Statement 5: $AD \parallel BC$ (By the same reasoning for $\angle ADB = \angle CDB$).

   • Reason 5: Alternate interior angles theorem.

6. Statement 6: Therefore, quadrilateral $ABCD$ is a parallelogram (since both pairs of opposite sides are parallel).

   • Reason 6: Definition of a parallelogram (if both pairs of opposite sides are parallel, the quadrilateral is a parallelogram).

Would you like further clarification on any of the steps or concepts? Here's a tip:

• The key to these types of proofs is understanding and applying postulates like SSS and geometric theorems like alternate interior angles.

Let me know if you need more assistance or additional questions to explore!

1. What is the importance of congruence in geometric proofs?
2. Can you list other methods for proving quadrilaterals are parallelograms?
3. What are alternate interior angles and how do they relate to parallel lines?
4. How does the SSS postulate work in proving triangle congruence?
5. What is the definition of a parallelogram and how can it be proven using other geometric properties?