The figure shows a parallelogram $ABCD$, with additional points $M$ on side $AB$ and $N$ on side $CD$. The lines $AM$ and $CN$ intersect inside the parallelogram at points $P$ and $Q$, forming the shaded quadrilateral $PMQN$. You are asking if the shaded area is a parallelogram.

To determine if the shaded region $PMQN$ is a parallelogram, let's consider the conditions for a quadrilateral to be a parallelogram:

• Opposite sides must be parallel.
• Opposite sides must be equal in length.

From the diagram:

1. Since $ABCD$ is already a parallelogram, the opposite sides $AB \parallel CD$ and $AD \parallel BC$.
2. Given that points $M$ and $N$ are midpoints, the segments $PM \parallel QN$ and $PQ \parallel MN$ should be parallel as well, because they form part of diagonals of the smaller triangles created by the lines $AM$ and $CN$.

This satisfies the key condition for $PMQN$ to be a parallelogram: opposite sides are parallel.

Conclusion:

Yes, the shaded region $PMQN$ is a parallelogram because both pairs of opposite sides appear to be parallel, given that $M$ and $N$ are midpoints and the structure of the larger parallelogram $ABCD$.

Would you like a deeper explanation of why the diagonals create parallel lines or have any questions?

Here are 5 related questions to expand your understanding:

1. What are the properties of a parallelogram that make it distinct from other quadrilaterals?
2. How can you prove that two lines are parallel in geometry?
3. What role do midpoints play in geometric figures like parallelograms?
4. How can you find the area of a parallelogram given its sides or diagonals?
5. What other shapes can be formed by intersecting diagonals inside a parallelogram?

Tip: Whenever a quadrilateral is formed by the intersection of diagonals in a larger parallelogram, it's often useful to check for parallel lines as a quick way to identify whether the smaller shape is also a parallelogram.