Tikzpicture: Add slope angles to straight line

Question

I am trying to add some angles between the x-axis and the red and blue line which indicate their slope.

\documentclass[15pt,a4paper,headinclude, totoc]{scrartcl}

\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}                                               
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\usetikzlibrary{calc}
\usetikzlibrary{intersections}
\usetikzlibrary{positioning}        
\usetikzlibrary{shapes}

\begin{document}
  \begin{center} 
\begin{tikzpicture}[scale=1.5]
\pgfplotsset{ticks=none}
  \begin{axis}[axis lines=middle,xmin=-0.2,xmax=4.5,ymin=-0.5,ymax=15, ultra thick,
    xlabel=$\scriptstyle C_t$,
    ylabel=$\scriptstyle C_{t+1}$,
    x label style={at={(axis description cs:1,-0.01)},anchor=north},
    y label style={at={(axis description cs:-0.15,1)},anchor=west},
        ]

        \addplot+[no marks,blue,domain=0:3.02,samples=200, thick] {-(9.6646)*x+29.12722};
        \addplot+[no marks,ForestGreen,domain=0:2.09,samples=200, thick] {-(9.6646)*x+20.15286};
        \addplot+[no marks,red,domain=0:3.02,samples=200, thick] {-(3.5799723)*x+10.78923};

        \addplot+[no marks,gray,domain=1.2:4,samples=200, thick] {(13.06666)/(0.8*x-0.4) -0.83333} node[right]{\footnotesize $U^0$};
        \addplot+[no marks,gray,domain=0.8:4,samples=200, thick] {(5.4)/(0.8*x-0.4) -0.83333 } node[right]{\footnotesize $U^1$};

        \fill (1.8,11.73094) circle (2pt) node[right]{\tiny \textbf{A}};
        \draw[gray, dash pattern=on 6pt off 3pt] (1.8,0) -- (1.8,11.73094);
        \draw[gray, dash pattern=on 6pt off 3pt] (0,011.73094) -- (1.8,11.73094);

        \fill[black] (1.8533,4.15433) circle (2pt) node[right]{\tiny \textbf{B}};
        \fill (1.3357,7.2435) circle (2pt) node[right]{\tiny \textbf{C}};

  \end{axis}
\end{tikzpicture}
\end{center}
\end{document}

Here you see my result so far. I need to adjust the labeling of the slopes. Thank you

Answer 1

Try

    \documentclass[15pt,a4paper,headinclude, totoc]{scrartcl}

    \usepackage[usenames,dvipsnames]{xcolor}
    \usepackage{tikz}
    \usepackage{pgfplots}
    \pgfplotsset{compat=1.12}
    \usetikzlibrary{angles,calc,intersections,quotes,positioning,shapes}

    \begin{document}
      \begin{center}
    \begin{tikzpicture}[scale=1.5,
my angle/.style args = {#1/#2}{draw,<->,
                               angle radius=#1,
                               angle eccentricity=#2,
                               } % angle label position!
                            ]
    \pgfplotsset{ticks=none}
      \begin{axis}[axis lines=middle,xmin=-0.2,xmax=4.5,ymin=-0.5,ymax=15, ultra thick,
        xlabel=$\scriptstyle C_t$,
        ylabel=$\scriptstyle C_{t+1}$,
        x label style={at={(1,-0.01)},anchor=north},
        y label style={at={(-0.15,1)},anchor=west},
       no marks,
       samples=100
            ]
        \coordinate (O) at (0,0);
    \addplot+[blue,domain=0:3.02,thick] {-(9.6646)*x+29.12722};
        \coordinate (a1) at (0,29.12722);
        \coordinate (a2) at (3.0138,0);
            \pic [my angle=12mm/1.2, "$\alpha$",blue] {angle = a1--a2--O};
    \addplot+[ForestGreen,domain=0:2.09,thick] {-(9.6646)*x+20.15286};
    \addplot+[red,domain=0:3.02,thick] {-(3.5799723)*x+10.78923};
        \coordinate (b1) at (0,20.15286);
        \coordinate (b2) at (2.08522,0);
            \pic [my angle=19mm/1.15, "$\beta$",red] {angle = b1--b2--O};

    \addplot+[gray,domain=1.2:4,thick] {(13.06666)/(0.8*x-0.4) -0.83333} node[right]{\footnotesize $U^0$};
    \addplot+[gray,domain=0.8:4,thick] {(5.4)/(0.8*x-0.4) -0.83333 } node[right]{\footnotesize $U^1$};

    \fill (1.8,11.73094) circle (2pt) node[right]{\tiny \textbf{A}};
    \draw[gray, dash pattern=on 6pt off 3pt] (1.8,0) -- (1.8,11.73094);
    \draw[gray, dash pattern=on 6pt off 3pt] (0,011.73094) -- (1.8,11.73094);

    \fill[black] (1.8533,4.15433) circle (2pt) node[right]{\tiny \textbf{B}};
    \fill (1.3357,7.2435) circle (2pt) node[right]{\tiny \textbf{C}};
        \end{axis}
    \end{tikzpicture}
    \end{center}
    \end{document}

Above MWE gives:

Some explanation: For drawing angles I use tikz library angles (as suggest Bordaigorl in his comment. For it I determine coordinates, where red and blue lines intersect coordinate axes. Angles as such are drawn with\pic [my angle=12mm/1.2, "$\alpha$",blue] {angle = a1--a2--O};. For labels of angles is used library `quotes".

Edit: With pgfplots v1.12 for coordinates is not necessary to use axis description cs: anymore. Since, all lines have the same number of samples and parameter no marks, both can be moved in axis preamble. With this MWE become more concise.