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.   
 


Case 1:

Given: AB ≅ BC ≅ CD ≅ DE ≅ EA
       ∠A ≅ ∠B ≅ ∠E  
        


 1. Draw BE, BD and EC
 2. AB ≅ BE                      (Given)
 3. ᐃABE is isosceles
 4. ∠ABE ≅ ∠AEB
 5. ∠ABC ≅ ∠AED                      (Given) 
 6. m∠ABC - m∠ABE = m∠AED - m∠AEB
 7. ∠CBE ≅ ∠DEB
 8. BE ≅ BE
 9. BC ≅ ED                          (Given)
10. ᐃBCE ≅ ᐃEDB                      SAS
11. BD ≅ EC
12. BC ≅ ED                          (Given)
13. CD ≅ CD
14. ᐃBCD ≅ ᐃEDC                     SSS 
15. ∠BCD ≅ ∠EDC

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

Given: AB ≅ BC ≅ CD ≅ DE ≅ EA
       ∠A ≅ ∠C ≅ ∠D 



 1. Draw BE, BD and EC
 2. ∠BCD ≅ ∠EDC               given
 3. BC ≅ DE                   given 
 4. CD ≅ CD                
 5. ᐃBCD ≅ ᐃEDC               SAS
 6. BD ≅ EC
 7. BC ≅ DE                   given
 8. BE ≅ BE
 9. ᐃBCE ≅ ᐃEDB
10. AB ≅ AE                   given
11. ᐃABE is isosceles
12. ∠ABE ≅ ∠AEB
13. ∠DBE ≅ ∠BEC
14. ∠CBD ≅ ∠DEC
15. m∠ABE + m∠DBE + m∠CBD = m∠AEB + m∠BEC + m∠DEC
16. ∠ABC ≅ ∠AED

Edwin