Point E is the midpoint of side BC of parallelogram ABCD, and point F is the midpoint of side

AD. Prove that quadrilateral BEDF is a parallelogram.

To prove that quadrilateral BEDF is a parallelogram, we need to show that its opposite sides are parallel.

Given that point E is the midpoint of side BC and point F is the midpoint of side AD, we can use several properties of parallelograms to prove that BEDF is also a parallelogram.

1. Opposite sides of a parallelogram are parallel. We will prove that BE is parallel to FD.

Since E is the midpoint of BC, we can say that BE is equal in length to EC. Similarly, F is the midpoint of AD, so AF is equal in length to FD.

Now, let's consider triangle ABC. Since it is a parallelogram, AB is parallel to CD. We can use this property to prove that BE is parallel to FD.

In triangle ABC, using the Midpoint Theorem, we can say that BE is equal in length to EC, and AF is equal in length to FD.

Using the Transitive Property of Equality, we can say that BE is equal in length to EC, which is equal to AF, which is equal to FD.

Therefore, by the Converse of the Midpoint Theorem, we can conclude that BE is parallel to FD.

2. Similarly, we can also prove that ED is parallel to BF.

Since E is the midpoint of BC, we can say that EC is equal in length to BE. Similarly, F is the midpoint of AD, so FD is equal in length to AF.

Using the same reasoning as in step 1, we can say that EC is equal in length to BE, which is equal to FD, which is equal to AF.

Therefore, by the Converse of the Midpoint Theorem, we can conclude that ED is parallel to BF.

By proving that both pairs of opposite sides, BE parallel to FD and ED parallel to BF, are parallel, we have shown that quadrilateral BEDF is a parallelogram.