Question 65860
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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.
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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°

 <font face = "symbol">Ð</font>G + <font face = "symbol">Ð</font>EIG + <font face = "symbol">Ð</font>GEI = 180°
 25° + <font face = "symbol">Ð</font>EIG + 62° = 180°
       87° + <font face = "symbol">Ð</font>EIG = 180°
             <font face = "symbol">Ð</font>EIG = 93°

 <font face = "symbol">Ð</font>V + <font face = "symbol">Ð</font>EIV + <font face = "symbol">Ð</font>IEV = 180°
100° +  37° + <font face = "symbol">Ð</font>IEV = 180°
      137° + <font face = "symbol">Ð</font>IEV = 180°
             <font face = "symbol">Ð</font>IEV = 43° 

In <font face = "symbol">D</font>EGI, <font face = "symbol">Ð</font>G < <font face = "symbol">Ð</font>GEI < <font face = "symbol">Ð</font>EIG, so their opposite sides
are respectively in the same order:

 EI < GI < EG

In <font face = "symbol">D</font>EIV, <font face = "symbol">Ð</font>EIV < <font face = "symbol">Ð</font>IEV < <font face = "symbol">Ð</font>V, 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</pre>