Question 571345
<pre>
There are two cases.  One case is when the three angles that are given
congruent are adjacent (the first figure below), and the other case is 
when they are not (the second figure below.   Mostly I will give only
the steps.  You must give the reasons.   
 
{{{drawing(300,300,-1.5,1.5,-1.5,1.5,

line(cos(18*pi/180),sin(18*pi/180),cos(90*pi/180),sin(90*pi/180)),
line(cos(90*pi/180),sin(90*pi/180),-cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)),
line(-cos(54*pi/180),-sin(54*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(54*pi/180),-sin(54*pi/180),cos(18*pi/180),sin(18*pi/180)),
locate(-.6,-.83,C),locate(.55,-.83,D),
locate(1,.35,E),locate(-.02,1.15,A), locate(-1.05,.35,B),
red(
arc(cos(18*pi/180),sin(18*pi/180),.8,-.8,144,252),
arc(-cos(18*pi/180),sin(18*pi/180),.8,-.8,288,396),

arc(cos(90*pi/180),sin(90*pi/180),.8,-.8,216,324)
)) )}}}{{{drawing(300,300,-1.5,1.5,-1.5,1.5,

line(cos(18*pi/180),sin(18*pi/180),cos(90*pi/180),sin(90*pi/180)),
line(cos(90*pi/180),sin(90*pi/180),-cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)),
line(-cos(54*pi/180),-sin(54*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(54*pi/180),-sin(54*pi/180),cos(18*pi/180),sin(18*pi/180)),
locate(-.6,-.83,C),locate(.55,-.83,D),
locate(1,.35,E),locate(-.02,1.15,A), locate(-1.05,.35,B),

red(

arc(cos(90*pi/180),sin(90*pi/180),.8,-.8,216,324),
arc(cos(54*pi/180),-sin(54*pi/180),.8,-.8,72,180),
arc(-cos(54*pi/180),-sin(54*pi/180),.8,-.8,0,108))
) )}}}

Case 1:

Given: AB &#8773; BC &#8773; CD &#8773; DE &#8773; EA
       &#8736;A &#8773; &#8736;B &#8773; &#8736;E  
        
{{{drawing(300,300,-1.5,1.5,-1.5,1.5,

line(cos(18*pi/180),sin(18*pi/180),cos(90*pi/180),sin(90*pi/180)),
line(cos(90*pi/180),sin(90*pi/180),-cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)),
line(-cos(54*pi/180),-sin(54*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(54*pi/180),-sin(54*pi/180),cos(18*pi/180),sin(18*pi/180)),
locate(-.6,-.83,C),locate(.55,-.83,D),
locate(1,.35,E),locate(-.02,1.15,A), locate(-1.05,.35,B),
red(
arc(cos(18*pi/180),sin(18*pi/180),.8,-.8,144,252),
arc(-cos(18*pi/180),sin(18*pi/180),.8,-.8,288,396),

arc(cos(90*pi/180),sin(90*pi/180),.8,-.8,216,324)
),
green(line(-cos(18*pi/180),sin(18*pi/180),cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)) )





 )}}}

 1. Draw BE, BD and EC
 2. AB &#8773; BE                      (Given)
 3. &#5123;ABE is isosceles
 4. &#8736;ABE &#8773; &#8736;AEB
 5. &#8736;ABC &#8773; &#8736;AED                      (Given) 
 6. m&#8736;ABC - m&#8736;ABE = m&#8736;AED - m&#8736;AEB
 7. &#8736;CBE &#8773; &#8736;DEB
 8. BE &#8773; BE
 9. BC &#8773; ED                          (Given)
10. &#5123;BCE &#8773; &#5123;EDB                      SAS
11. BD &#8773; EC
12. BC &#8773; ED                          (Given)
13. CD &#8773; CD
14. &#5123;BCD &#8773; &#5123;EDC                     SSS 
15. &#8736;BCD &#8773; &#8736;EDC

-------------------------------
Case 2.

Given: AB &#8773; BC &#8773; CD &#8773; DE &#8773; EA
       &#8736;A &#8773; &#8736;C &#8773; &#8736;D 

{{{drawing(300,300,-1.5,1.5,-1.5,1.5,

line(cos(18*pi/180),sin(18*pi/180),cos(90*pi/180),sin(90*pi/180)),
line(cos(90*pi/180),sin(90*pi/180),-cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)),
line(-cos(54*pi/180),-sin(54*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(54*pi/180),-sin(54*pi/180),cos(18*pi/180),sin(18*pi/180)),
locate(-.6,-.83,C),locate(.55,-.83,D),
locate(1,.35,E),locate(-.02,1.15,A), locate(-1.05,.35,B),

red(

arc(cos(90*pi/180),sin(90*pi/180),.8,-.8,216,324),
arc(cos(54*pi/180),-sin(54*pi/180),.8,-.8,72,180),
arc(-cos(54*pi/180),-sin(54*pi/180),.8,-.8,0,108))
,
green(line(-cos(18*pi/180),sin(18*pi/180),cos(18*pi/180),sin(18*pi/180)),
line(-cos(18*pi/180),sin(18*pi/180),cos(54*pi/180),-sin(54*pi/180)),
line(cos(18*pi/180),sin(18*pi/180),-cos(54*pi/180),-sin(54*pi/180)) )




 )}}}

 1. Draw BE, BD and EC
 2. &#8736;BCD &#8773; &#8736;EDC               given
 3. BC &#8773; DE                   given 
 4. CD &#8773; CD                
 5. &#5123;BCD &#8773; &#5123;EDC               SAS
 6. BD &#8773; EC
 7. BC &#8773; DE                   given
 8. BE &#8773; BE
 9. &#5123;BCE &#8773; &#5123;EDB
10. AB &#8773; AE                   given
11. &#5123;ABE is isosceles
12. &#8736;ABE &#8773; &#8736;AEB
13. &#8736;DBE &#8773; &#8736;BEC
14. &#8736;CBD &#8773; &#8736;DEC
15. m&#8736;ABE + m&#8736;DBE + m&#8736;CBD = m&#8736;AEB + m&#8736;BEC + m&#8736;DEC
16. &#8736;ABC &#8773; &#8736;AED

Edwin</pre>