ABCD is a square and BCE is an equilateral triangle. Find the value of a, b and reflex angle BED.

Dear Student,

Please find below the solution to the asked query:

Given :  ABCD is a square , So

$\angle$ ABC  =  $\angle$ BCD = $\angle$ CDA = $\angle$ DAB  = 90$°$                                      --- ( 1 )

And

BCE  is a equilateral triangle , So

$\angle$ EBC  =  $\angle$ ECB = $\angle$ BEC = 60$°$                                                       --- ( 2 )

So,


$\angle$ DCE  =  $\angle$ BCD - $\angle$ ECB =  90$°$ - 60$°$  = 30$°$  ( Substitute value from equation 1 and 2 ) 

As given BCE is a equilateral triangle , SO BC  =  EC  =  EB   and ( AB = BC  =  CD  =  DA , Sides of square ) , So

EC  =  CD , Then triangle DEC is a isosceles triangle so from base angle theorem we get

$\angle$ CDE  =  $\angle$ CED                                                     --- ( 3 ) 

From angle sum property of triangle we get in triangle DEC :

$\angle$ CDE + $\angle$ CED + $\angle$  DCE =  180$°$ , Substitute value from equation 3 we get 

$\angle$ CED  + $\angle$ CED + $\angle$  DCE =  180$°$ ,  Now substitute values we get

b + b + 30$°$ =  180$°$

2 b = 150$°$

b = 75$°$ , So

$\angle$ CDE  =  $\angle$ CED = b = 75$°$ 

Then ,

a  =  $\angle$ CDA - $\angle$ CDE =  90$°$ - 75$°$  = 15$°$  ( Substitute value from equation 1 and $\angle$ CDE = 75$°$  ) 

And

Reflex $\angle$ BED  =  360$°$ - $\angle$ BEC - $\angle$CED =  360$°$ - 60$°$ - 75$°$ = 225$°$ .

therefore,

a = 15$°$ , b = 75$°$  and Reflex $\angle$ BED  =  225$°$                                                          ( Ans )


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