State and prove the Midsegment Theorem.

The correct answer is: Hence proved

Complete step by step solution:
 Triangle midsegment theorem states that the line segment connecting the midpoints
of any 2 sides of a triangle is

1. Is one half the length of the third side.

b. Is parallel to the third side
Proof:

In  , we connect 2 midpoints D and E of 2 sides AB and BC.
Consider 2 triangles,  
We have,
 (since D is the midpoint)
 (since E is the midpoint)
by SAS similarity criterion.
  are similar triangles.
We know that corresponding sides of similar triangles are proportional.

Here, we have proved the first part.
Now, Take 2 line segments DE and AC and a transversal BC cutting these 2 lines.
Since,    are similar triangles we have congruent corresponding
angles.


So by the converse of corresponding angles theorem, since a pair of corresponding
angles created by the transversal BC are congruent, we conclude that DE and AC
are parallel.
Hence proved.