Chapter 5.3, Problem 19WE

To determine

To prove: $DE=BC$

Explanation of Solution

Given information: ABCD is a parallelogram, $\overline{BE}\parallel \overline{MD}$ and M is the midpoint of $\overline{AB}$

Proof:

  

In $\Delta ABE$ , $\overline{MD}\parallel \overline{BE}$ and M is the midpoint of $\overline{AB}$

So,

D is a midpoint of $\overline{AE}$

(According to converse of midpoint theorem, if a line is drawn through the midpoint of a side of a triangle parallel to the other side, then it bisects the third side)

As D is a midpoint of $\overline{AE}$ ,

  $\overline{AD}=\overline{DE}(i)$

Now,

ABCD is a parallelogram.

So,

  $\overline{AD}=\overline{BC}(ii)$ (Opposite sides of a parallelogram are equal)

From $(i)\&(ii)$ ,

  $\overline{DE}=\overline{BC}$