Please solve the 19th problem.
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Q.19.  The following figure shows a triangle ABC with exterior angles as x, y and z.


(i) If AB > AC > BC; arrange the angles x, y and z in ascending order of their values.
(ii) In the same figure, if y > x > z; arrange sides AB, BC and AC in descending order of their lengths.

Dear Student,

Please find below the solution to the asked query:

Here , Exterior BAC + BAC = 180°                         ( Linear pair angles )
So,

x  + BAC = 180°   , By substituting value from given diagram

BAC  = 180° - x 

Similarly we get : ABC  = 180° - y   and ACB  = 180° - z 

Then ,

BAC  = 180° - x  , ABC  = 180° - y   and ACB  = 180° - z                      --- ( 1 )

i ) Given :  AB > AC > BC

And we know theorem : In two unequal sides of triangle , the angle opposite to longer side is larger vice versa . 

So ,

ACB > ABC >  BAC , Substitute values from equation 1 we get :

180° - z > 180° - y  > 180° - x  , Now we subtract 180° and get :

- z > - y > - x  , Now we multiply by ' - 1 ' and get :

z  <  y < x                                                                       ( Ans )

ii ) Given : y > xz  , Now we multiply by ' - 1 ' and get :

- y < - x  < - , Now we add 180° and get :

180°- y < 180°- x  < 180°- , Substitute values from equation 1 we get :

ABC <  BAC < ACB , By using above mention theorem we get :

AC < BC < AB

Then we write it in descending order , As :

AB > BC > AC                                                        ( Ans )


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