Question

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. B = 137°, c = 9, b = 14

Answer 1

Yes; a triangle is formed:
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m∡A = 17° ;
m∡B = 137° (given) ;
m∡C = 26° ;
   a = 6 ;
   b = 14 (given) ;
   c = 9 (given) .
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Explanation:
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Note:

The law of sines:

(sin A) / a = (sin B) / b = (sin C) / c ;
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Given: 
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B = 137 ;
c = 9 ;
b =14 ;
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 (sin B) / b = (sin C) / c ;
                
 (sin 137) / 14 = (sin C) / 9  ;

 (0.681998360062) / 14 = 0.0487141685758571 = (sin C) / 9  ;
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sin C = (0.0487141685758571) * (9) ;

sin C =  0.4384275171827139 ;

        Take the "arc sin" of each side of the equation; to isolate "C" on one side of the equation; and to solve for "C" ;

arc sin (C) = arc sin ( 0.4384275171827139) ;

           C = 26.003593520741  ; round to 26.

If all angles of a triangle add up to 180 degrees: then:

A + B + C = 180 ;

A + 137 + 26  = 180 ;

A + 163 = 180 ;

Subtract "163" from each side of the equation; to isolate "A" on one side of the equation; and to solve for "A" ;

A + 163 − 163 = 180 − 163  ;

      A = 17 ;

Now, to solve for "a" ;

(sin A) / a = (sin B)/ b ;

(sin 17) / a = 0.0487141685758571 ;

(0.0487141685758571) a = (sin 17) ;

Divide EACH SIDE of the equation by:  "(0.0487141685758571)"  ;
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           to isolate "a" on one side of the equation; and to solve for "a" ;
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[(0.0487141685758571)a ] / (0.0487141685758571) = 

                   (sin 17) / (0.0487141685758571) ;
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                   a =  (sin 17) / (0.0487141685758571) ;

                      = (0.292371704723) / (0.0487141685758571) ;

                   a = 6.0017796314787227112 ;  round to "6" .
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