Hi, I'm having a difficult time figuring this problem 
out. THANKS FOR HELPING!!

Find the longest and shortest line segment drawn in
the following diagram.


The sides of a triangle are always in the same order 
of size as the angles opposite them:

First we calculate the two angles we aren't given by 
using the fact that the three angles of any triangle 
must always have sum 180°

 ÐG + ÐEIG + ÐGEI = 180°
 25° + ÐEIG + 62° = 180°
       87° + ÐEIG = 180°
             ÐEIG = 93°

 ÐV + ÐEIV + ÐIEV = 180°
100° +  37° + ÐIEV = 180°
      137° + ÐIEV = 180°
             ÐIEV = 43° 

In DEGI, ÐG < ÐGEI < ÐEIG, so their opposite sides
are respectively in the same order:

 EI < GI < EG

In DEIV, ÐEIV < ÐIEV < ÐV, so their opposite sides
are respectively in the same order:

 EV < IV < EI

So we can put those two inequalities together, and
get

 EV < IV < EI < GI < EG

Therefore EV is the shortest side and EG is the 
longest side.

Edwin