Question 1117949
We can draw side {{{AB}}} , with length {{{c=14}}},
and draw a ray (arrow) that will contain side {{{AC}}} ,
starting from {{{A}}} and making an angle {{{A=55^o}}} with side {{{AB}}} .
We know that {{{C}}} is at a distance {{{a=13}}} from {{{B}}},
so it is going to be where the circle with center {{{B}}} and radius {{{13}}} crosses the ray (arrow) .
{{{drawing(300,300,-1,15,-1,15,
line(0,0,14,0),arrow(0,0,10,14.28),
green(arc(14,0,26,26,180,280)),
green(arrow(14,6.5,14,0)),
green(arrow(14,6.5,14,13)),
red(arc(0,0,6,6,-55,0)),
locate(2.6,2.6,red(A=55^o)),
locate(6.5,0,c=14),
locate(14,7,green(13)),
locate(-0.2,0,A),locate(13.8,0,B),
circle(1.09,1.56,0.2),circle(8.12,11.59,0.2),
locate(0.1,2,"C ?"),locate(8,11.35,"C ?")
)}}}
The problem is that in this case, we find two possible locations for {{{C}}} ,
so  the given measurements produce two​ triangles.
From the drawing above, we can see that if the measurement for side {{{BC}}} was {{{a=15}}} ,
we would have one​ triangle,
and if {{{BC}}} was very short (as in {{{a=5}}} for example),
we would have no triangle at all.
 
As we have measurements for one angle and the opposite side, {{{system(A=55^o,a=13)}}} ,
we can solve for the two​ triangles using Law of Sines:
{{{sin(A)/a=sin(B)/b=sin(C)/c}}} or {{{a/sin(A)=b/sin(B)=c/sin(C)}}} .
 
Having {{{A}}} , {{{a}}} , and {{{c}}} , we start by using
{{{sin(A)/a=sin(C)/c}}} to try to find {{{C}}}:
{{{0.819152/13=sin(C)/14}}}
{{{14*0.819152/13=sin(C)}}}
{{{sin(C)=0.882164}}}
My calculator says that {{{C=61.9^o}}} is a solution.
That gives us
TRIANGLE #1, with
{{{B=180^o-(A+C)=180^o-(55^o+61.9^o)=180^o-116.9^o=63.1^o}}} .
Now that we have {{{B}}} , we can use {{{a/sin(A)=b/sin(B)}}} to find {{{b}}} .
{{{13/0.819152=b/sin(63.1^o)}}}
{{{13/0.819152=b/0.891798}}}
{{{13*0.891798/0.819152=b}}}
{{{b=14.15}}}
 
TRIANGLE #2:
However, {{{C=180^o-61.9^o=118.1^o}}} also has {{{sin(C)=0.882164}}} ,
and a triangle with {{{A=55^o}}} and {{{C=118.1^o}}} is also possible,
because {{{A+C=55^o+118.1^o}}} is less than {{{180^o}}} .
That triangle would have {{{C=118.1^o}}} and
{{{B=180^o-(A+C)=180^o-(55^o+118.1^o)=180^o-173.1^o=6.9^o}}} .
With {{{B=6.9^o}}} , {{{a/sin(A)=b/sin(B)}}} gives us
{{{13/0.819152=b/sin(6.9^o)}}}
{{{13/0.819152=b/0.120137}}}
{{{13*0.1201378/0.819152=b}}}
{{{b=1.91}}} .