In triangle ABC. if angle ABD = 64, what is the measure of angle DEC - angle ACE

Q. In triangle ABC. if ​ $\text{∠}$ ABD = 64, what is the measure of   $\text{∠}$ DEC - $\text{∠}$ACE

Dear Student,

Please find below the solution to the asked query:

Given :  $\angle$ BAC  = 90$°$  and $\angle$ BDA  = 90$°$ and $\angle$ CDA  = 90$°$ , from diagram , $\angle$ ABD  = 64$°$ also given . 

From angle property of triangle we know sum of all internal angles are 180$°$ ,

In $∆$BAD we get
$\angle$ BDA + $\angle$ BAD  + $\angle$ ABD  =  180$°$ , Substitute given values we get

90$°$ + $\angle$ BAD + 64$°$ = 180$°$

$\angle$ BAD  + 154$°$ =  180$°$

$\angle$ BAD = 26$°$

And we know :

$\angle$ BAC  =  $\angle$ BAD + $\angle$ CAE , So

90$°$ = 26$°$  + $\angle$ CAE

$\angle$ CAE = 64$°$

We know from external angle theorem in $∆$ ACE :

$\angle$ CAE + $\angle$ ACE  =  $\angle$ DEC , Substitute value and get

64$°$ + $\angle$ ACE  =  $\angle$ DEC ,

$\angle$ DEC  -  $\angle$ ACE  = 64$°$                                                                        ( Ans )

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