You just need to calculate the proper coordinates, for example
\join(0.408,0.707)(0.4513,0.682)(0.4763,0.7253)
\join(0.707,0.408)(0.732,0.3647)(0.775,0.3897)
How did I get to those values? From the one you had already drawn, you know that the length of a line from the origin to the start of the angle mark, is 0.816, and that the sides of the angle mark has lengths 0.05.
The second ray has the angle 60 degrees with the horizontal, so if you want the angle mark at the same position along the ray, it should start at the coordinate (0.816*cos(60), 0.816*sin(60)). The next thing you need to find is the vector describing the lower side of the angle mark, which is (0.05*sin(60), -0.05*cos(60)). Add this vector to the start point of the angle mark, and you have the corner. I'm sure you can figure then how I got the last point.
That said, there are other packages for making diagrams. Below I included one example of how you can make that diagram using TikZ.
\documentclass{article}
\usepackage{2130}
\usepackage{tikz}
\usetikzlibrary{
decorations.pathreplacing,
intersections,
calc
}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[
declare function={
R=5;
}
]
% define a named coordinate at origin
\coordinate (O) at (0,0);
% draw the arc
\draw (O) ++(R,0) arc[start angle=0,end angle=125,radius=R];
% draw the three rays
\foreach \angle in {30,60,90}
{
% line from origin to arcm using polar coordinates (the ":" indicates that the coordinate is (angle:radius).)
% at the point where the ray hits the circle, a named coordinate is added
\draw (O) --node[auto]{$R$} ++(\angle:R) coordinate (r\angle);
% draw right angle marks, again with polar coordinates
\draw (\angle:R*0.95) -- ++(\angle-90:R*0.05) -- ++(\angle:R*0.05);
}
% the two next paths doesn't draw anything, but are used to find the intersection of the tangents
\path [overlay,name path=A] (r90) -- ++(R,0);
\path [overlay,name path=B] (r60) -- ++(60:R);
% find the intersection of the two paths above, name the coordinate "z"
% draw the two tangents starting at the point z
\draw [name intersections={of=A and B,by={z}}]
(z) -- ++(-1.2*R,0)
(z) -- ++(-60:1.2*R);
% draw the extension of the middle ray
\draw [dashed] (r60) -- node[left,pos=0.7,font=\tiny] {$Z$} (z);
% draw the dashed Y-lines
% the sine law is used to calculate the length of the lines
\draw [densely dashed]
let
\p1=(z),\p2=(r60),\n1={veclen(\x1-\x2,\y1-\y2)*sin(60)/sin(30)}
in
(r60) -- node[above,pos=0.4,font=\tiny] {$Y$} ++(60+90:\n1) coordinate (y1)
(r60) -- node[above,pos=0.4,font=\tiny] {$Y$} ++(60-90:\n1) coordinate (y2);
% draw the braces on top
\draw [decoration={brace,raise=1pt,amplitude=3pt},decorate,thick] (r90) -- node[above=2pt,font=\footnotesize]{$Y$} (y1);
\draw [decoration={brace,raise=15pt,amplitude=3pt},decorate,thick] (r90) -- node[above=16pt,font=\footnotesize]{$X$} (z);
% draw black dot in origin
\fill (O) circle[radius=2pt];
\end{tikzpicture}
\caption{Recurrence Relation, Circumscribed Polygon}
\label{descriptivelabel}
\end{figure}
\begin{figure}
\begin{scaledpicture}{50}(1,1)(-0.3,0)
\put(0,.866){\makebox(0.5,0.18)\downbracefill}
\put(0,.866){\makebox(0.25,0.045)\downbracefill}
\put(0,0){\circle*{0.03}}
\join(0,.866)(0.05,.866)(0.05,0.816)(0,.816)
%
\join(0.408,0.707)(0.4513,0.682)(0.4763,0.7253)
\join(0.707,0.408)(0.732,0.3647)(0.775,0.3897)
%
\join(0.408,0.707)(0.4513,0.682)(0.4763,0.7253)%
\put(0,0){\arc(0.866,0){125}}
\join(0,0)(0.433,0.750)
\dashline{0.03}(0.433,0.750)(0.5,0.866)
\join(0,0)(0,0.866)
\join(0,0)(0.745,0.433)
\join(.6225,.6495)(.745,.433)
\dashline{0.01}(0.25,0.866)(.6225,.6495)
\thicklines \join(-0.5,0.866)(0,.866)
\thicklines \join(0.75,.433)(1,0)
\thicklines \join(0.745,.433)(0.5,0.866)(0,.866)
\wput(0,0.5){$R$}
\nput(0.3725,0.2165){$R$}
\nwput(.2165,.3725){$R$}
\nput(0.25,0.96){$X$}
\nput(0.135,0.888){$Y$}
\tiny \put(.52,.71){$Y$}
\wput(.475,.83){$Z$}
\nput(0.36,.78){$Y$}
\end{scaledpicture}
\caption{Recurrence Relation, Circumscribed Polygon}
\label{descriptivelabel}
\end{figure}
\end{document}
