In the figure, point $A$ is the center of the circle, the measure of angle $RAS$ is 74 degrees, and the measure of angle $RTB$ is 28 degrees. What is the measure of minor arc $BR$, in degrees? [asy]
unitsize(1.2cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=3;

pair A=(0,0), B=(-1,0), T=(2,0);
pair T0=T+10*dir(162);
pair[] RS=intersectionpoints(Circle(A,1),T--T0);
pair Sp=RS[0];
pair R=RS[1];
pair[] dots={A,B,T,Sp,R};
dot(dots);

draw(Circle(A,1));
draw(B--T--R);

label("$T$",T,S);
label("$A$",A,S);
label("$B$",B,W);
label("$R$",R,NW);
label("$S$",Sp,NE);
[/asy]
Let $C$ be the point where line segment $\overline{AT}$ intersects the circle.  The measure of $\angle RTB$ half the difference of the two arcs it cuts off: \[
m \angle RTB = \frac{m\widehat{RB}-m\widehat{SC}}{2}.
\] Since $m\widehat{RS}=74^\circ$, $m\widehat{SC}=180^\circ-74^\circ-m\widehat{RB}$.  Substituting this expression for $m\widehat{SC}$ as well as $28^\circ$ for $m \angle RTB$, we get \[
28^\circ = \frac{m\widehat{RB}-(180^\circ-74^\circ-m\widehat{RB})}{2}.
\] Solve to find $m\widehat{RB}=\boxed{81}$ degrees.

[asy]
unitsize(1.2cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=3;
pair A=(0,0), B=(-1,0), T=(2,0), C=(1,0);
pair T0=T+10*dir(162);
pair[] RS=intersectionpoints(Circle(A,1),T--T0);
pair Sp=RS[0];
pair R=RS[1];
pair[] dots={A,B,T,Sp,R,C};
dot(dots);
draw(Circle(A,1));
draw(B--T--R);
label("$T$",T,S);
label("$A$",A,S);
label("$B$",B,W);
label("$R$",R,NW);
label("$S$",Sp,NE);
label("$C$",C,SE);[/asy]