#### Answer

1. The edges of trapezoids are straight lines.
2. Thus, as a line in between the two sides of the trapezoids goes downward, its length decreases constantly.
3. Because its length decreases constantly, its length will be the average of the upper length and the bottom length. Using the equation for average, we obtain:
$EF = 1/2(AB + DC)$

#### Work Step by Step

Using theorem 4.4.7 we conclude that segment AM = MC. So that M is the midpoint of segment AC.
1- in a triangle ADC, EM= 1/2 DC
2- in a triangle ABC, MF= 1/2 AB
3- by segment addition property
EF= EM+ MF therefore by substitution EF= 1/2 DC+ 1/2 AB .