Let’s analyze the problem provided in the image.

The question asks us to prove that Quadrilateral ABCD is a parallelogram, given that:

1. $AD \cong BC$ (AD is congruent to BC, meaning they have equal lengths).
2. $AD \parallel BC$ (AD is parallel to BC).

Solution:

To prove that Quadrilateral ABCD is a parallelogram, we rely on the following theorem:

• A quadrilateral is a parallelogram if one pair of opposite sides is both equal in length and parallel.

Proof:

1. From the problem, we know $AD \parallel BC$. This satisfies the condition of parallelism for opposite sides.
2. Additionally, $AD \cong BC$, satisfying the condition of equality for opposite sides.

Since one pair of opposite sides ($AD$ and $BC$) is both parallel and equal, Quadrilateral ABCD is a parallelogram by definition.

Would you like further clarification on this solution or want me to expand on any part of it?

Related Questions:

1. What are the other ways to prove a quadrilateral is a parallelogram?
2. How do you calculate the slopes of sides to verify parallelism in a coordinate geometry setting?
3. Can a quadrilateral still be a parallelogram if one pair of opposite sides is equal, but not parallel?
4. What is the relationship between the diagonals in a parallelogram?
5. How do the properties of angles in a parallelogram support this proof?

Tip:

When working with geometric proofs, always refer to theorems and definitions to build logical arguments for your conclusions.