Finding Angle Measures

First of all thanks for attempting the problem.

Solution -

Since it is a regular polygon (which means all angles are equal in measure & all sides have the same length)
∠A=∠B=∠C=∠D=∠E=160°

1. Finding the value of x

      Proof :      ∠BCD=160° {proved above}
                        ∠MCD=90° {since CDLM is a square}
   

   

    So,  
           X + ∠MCD = ∠BCD
           X + 90° = 160°
           X = 160° - 90°
           X = 70°
   

2. Finding the value of y

      Proof :     
                      Since,  AB=BC=CD=DE {proved above}
                      Also,    CD=CM {because CDLM is a square}
                      Therefore, CB=CM which means CBM is an Isosceles Triangle
   
      Now, 
                As we know that Isosceles Triangle has two equal sides and angles 
                  ∠CBM = ∠CMB
      So, 
                X + ∠CBM + ∠CMB = 180°
               70° + ∠CBM + ∠CBM = 180° { Since ∠CBM =∠ CMB }
               70° + 2∠CBM = 180°
               2∠CBM = 180° - 70°
               ∠CBM = 110/2
               ∠CBM = 55°
   
     Therefore, 
                         Y + ∠CBM = ∠ABC
                         Y  + 55° = 160°
                         Y = 160° - 55°
                         Y = 105°
   

3. Adding both the values

       We get,      
                          X + Y 
                          70° + 105° = 175°

Hence Proved.................

Since CDLF is a square, CF + CD + BC since the polygon is regular, and triangle BCF is isosceles < BCF is 70, since interior angles are 160 and <FCB =90. Since triangle BCF is isosceles <CBF = <CFB = (1/2)*(180 - 70) = 55. Therefore x + y = 70 +(160 -55) = 175. Ed Gray