Final answer:
To prove quadrilateral BEDF is a parallelogram, we show that opposite sides are congruent and parallel, utilizing the properties of parallelogram ABCD and the fact that E and F are midpoints of BC and AD, respectively.
Step-by-step explanation:
The question is asking to prove that quadrilateral BEDF is a parallelogram, given that point E is the midpoint of side BC and point F is the midpoint of side AD of parallelogram ABCD. To prove this, we need to show that in quadrilateral BEDF, opposite sides are congruent and parallel.
Since ABCD is a parallelogram, AB is parallel and congruent to CD, and AD is parallel and congruent to BC. Because E and F are midpoints, AE is congruent to EB, and CF is congruent to FD. Therefore, BEDF has one pair of opposite sides that are both parallel and congruent (BE || DF and BE ≅ DF because they are both half of BC and AD respectively).
Additionally, DE and BF are congruent because DE is half of CD and BF is half of AB, and these sides are parallel and congruent in parallelogram ABCD. This gives us a second pair of opposite sides in BEDF that are parallel and congruent. Thus, by the definition of a parallelogram, which states that a quadrilateral with both pairs of opposite sides parallel and congruent is a parallelogram, BEDF is a parallelogram.