How can I draw angles only with the outer line, but not with the line to the center?

Question

I have the following LaTeX-Code:

\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}
\usepackage{tikz}
\usetikzlibrary{shapes, calc, shapes,snakes} 
\usepackage{amsmath,amssymb}

\begin{document}
\begin{preview}
\begin{tikzpicture}[thick]
    \newcommand\XC{4}
    \newcommand\ALPHA{30}

    \newcommand\XB{{(\XC*(cos(\ALPHA)*cos(\ALPHA) - sin(\ALPHA)*sin(\ALPHA)) + \XC)*0.5}}
    \newcommand\YB{{sqrt(cos(\ALPHA)*\XC*cos(\ALPHA)*\XC - (\XC*(cos(\ALPHA)*cos(\ALPHA) - sin(\ALPHA)*sin(\ALPHA)) + \XC)*0.5*(\XC*(cos(\ALPHA)*cos(\ALPHA) - sin(\ALPHA)*sin(\ALPHA)) + \XC)*0.5)}}

    \newcommand\XBo{{(\XB)+4}}
    \newcommand\YBo{{\YB}}
    \newcommand\XBp{{\XB*0.5}}
    \newcommand\YBp{{\YB*0.5}}

    % Draw the triangle
    \draw[fill=gray!10]  (0,    0) coordinate (A) 
        -- (\XB,\YB) coordinate (C) 
        -- (\XC,  0) coordinate (B) 
        -- (0,    0);

    % Draw nodes
    \node at (A)[anchor=north] {A};
    \node at (B)[anchor=north] {B};
    \node at (C)[anchor=south] {C};

    % Draw edge text
    \node (c) at ($(A)!0.5!(B)$) [below] {c};
    \node (b) at ($(A)!0.5!(C)$) [above] {b};
    \node (a) at ($(B)!0.6!(C)$) [right] {a};


    % draw alpha
    \draw[fill=green!30] (0,0) -- (0:0.75cm) arc (0:\ALPHA:.75cm);
    \coordinate[label=right:$\alpha$] (Alpha) at (0.25,0.15);

    % draw beta
    \begin{scope}[shift={(\XC, 0)}]
        \draw[fill=red!30] (0,0) -- (-180:0.50cm) arc (180:{180-(90-\ALPHA)}:0.5cm);
        \draw (150:0.35cm) node {$\beta$};
    \end{scope}

    % draw gamma
    \begin{scope}[shift={(\XB, \YB)}]
        \draw[fill=blue!30] (0,0) -- ({180+\ALPHA}:0.5cm) arc ({180+\ALPHA}:{180+\ALPHA+90}:0.5cm);
        \draw (-0.1, -0.05) node[below] {$\gamma$};
    \end{scope}


    % shifted alpha
    \begin{scope}[shift={(4,0)}]
        \draw[fill=green!30] (0,0) -- (0:0.5cm) arc (0:\ALPHA:.5cm);
    \end{scope}

    % shifted gamma
    \begin{scope}[shift={(4, 0)},rotate=180]
        \draw[fill=blue!30] (0,0) -- ({180+\ALPHA}:0.5cm) arc ({180+\ALPHA}:{180+\ALPHA+90}:0.5cm);
    \end{scope}

    \begin{scope}[shift={(3.13,-0.5)}]
        \draw[dashed,red]  (0,    0)  -- (\XB,\YB);
    \end{scope}

    \draw[dashed, color=red] (-1, 0) -- (6, 0);
    \draw  (0,    0) coordinate (X) 
        -- (\XB,\YB) coordinate (Y) 
        -- (\XC,  0) coordinate (Z) 
        -- (0,    0);
\end{tikzpicture}
\end{preview}
\end{document}

That produces:

When you look close at the horizontal red dashed line, you can see a black line from the green angle. I would like to get rid of it without changing anything else in the image

I know, it comes from this line:

\draw[fill=green!30] (0,0) -- (0:0.5cm) arc (0:\ALPHA:.5cm);

I could get rid of the line, if I changed the code to this:

\draw[fill=green!30] (0:0.5cm) arc (0:\ALPHA:.5cm);

But then I get this:

The green background disappeared. How can I get green background, but not the black line?