TikZ using \tkzMarkAngle in pics

Do you want something like this?

\documentclass[border=10pt,tikz,multi]{standalone}
\tikzset{%
  pics/dart/.style = {%
    code = {%
      \draw [pic actions] (0,0) coordinate(o1)
      -- (0:\PhiLen) coordinate (o2)
      -- ([turn]144:\Len) coordinate (o3)
      -- ([turn]-36:\Len) coordinate (o4)
      -- cycle;
      \draw[dashed, pic actions] (o1) -- (o3);
      \draw[green, pic actions] (o1) +(\OverPhiLen,0) arc (0:72:\OverPhiLen);
      \draw[red, pic actions] (o3) ++(-36:\OverPhiLen)  arc (288:144:\OverPhiLen);
    }
  }
}
\begin{document}
\begin{tikzpicture}
  \pgfmathsetlengthmacro{\Len}{5cm};
  \pgfmathsetlengthmacro{\Phi}{1.6180339};
  \pgfmathsetlengthmacro{\PhiLen}{\Len * \Phi };
  \pgfmathsetlengthmacro{\OverPhiLen}{\Len / \Phi };
  \pgfmathsetlengthmacro{\BigAngLen}{ \Len / \Phi};
  \pgfmathsetlengthmacro{\SmallAngLen}{ \BigAngLen / \Phi};
  \foreach \s in {0,72,144,216,288}
  \draw (1,0) pic [rotate=\s] {dart};
\end{tikzpicture}
\end{document}

And here's a version which labels the angles by adding arrows and numbers. I'm not sure if these are the angles you want marked, but presumably the idea could be modified as desired. Here's the idea:

\documentclass[border=10pt,tikz,multi]{standalone}
\usetikzlibrary{arrows.meta,bending}
\begin{document}
\tikzset{%
  dart label/.style={%
    midway, fill=white
  },
  pics/dart/.style = {%
    code = {%
      \draw [pic actions] (0,0) coordinate(o1)
      -- (0:\PhiLen) coordinate (o2)
      -- ([turn]144:\Len) coordinate (o3)
      -- ([turn]-36:\Len) coordinate (o4)
      -- cycle;
      \draw [dashed, pic actions] (o1) -- (o3);
      \draw [green!75!black,  -{Stealth[bend]}, pic actions] (o1) +(\OverPhiLen,0) arc (0:72:\OverPhiLen);
      \path [green!75!black, pic actions] (o1) +(.75*\OverPhiLen,0) arc (0:72:.75*\OverPhiLen) node [dart label] {72};
      \draw [red,  -{Stealth[bend]}, pic actions] (o3) ++(-36:\SmallAngLen)  arc (-36:-252:\SmallAngLen);
      \path [red, pic actions] (o3) ++(-36:.5*\SmallAngLen)  arc (-36:-252:.5*\SmallAngLen) node [dart label] {216};
      \draw [blue,  -{Stealth[bend]}, pic actions] (o3) ++(-36:.75*\SmallAngLen)  arc (-36:108:.75*\SmallAngLen);
      \path [blue, pic actions] (o3) ++(-36:.5*\SmallAngLen)  arc (-36:108:.5*\SmallAngLen) node [dart label] {144};
    }
  }
}
\begin{tikzpicture}
  \pgfmathsetlengthmacro{\Len}{5cm};
  \pgfmathsetlengthmacro{\Phi}{1.6180339};
  \pgfmathsetlengthmacro{\PhiLen}{\Len * \Phi };
  \pgfmathsetlengthmacro{\OverPhiLen}{\Len / \Phi };
  \pgfmathsetlengthmacro{\BigAngLen}{ \Len / \Phi};
  \pgfmathsetlengthmacro{\SmallAngLen}{ \BigAngLen / \Phi};
  \foreach \s in {0,72,144,216,288}
  \draw (1,0) pic [rotate=\s] {dart};
\end{tikzpicture}
\end{document}

One may change sqrt(2)-1 bye whatever between 0 and 1.

\documentclass[tikz,border=4]{standalone}
\usetikzlibrary{through,calc,angles}
\usepackage{siunitx}

\tikzset{
pics/dart/.style = {
    code ={ %
    \begin{scope}
    \clip[draw] (0,0)       coordinate (O1)
       -- (0:\PhiLen)       coordinate (O2)
       -- ([turn]+144:\Len) coordinate (O3)
       -- ([turn]-36:\Len)  coordinate (O4)
       -- cycle;          
    \draw[dashed] (O1) -- (O3);
    \coordinate (O5) at ($(O1)!sqrt(2)-1!(O3)$) ;
    \coordinate (O6) at ($(O1)!2/3!(O3)$) ;
    \node [draw,green] at (O1) [circle through=(O5)] {} ;
    \node [draw,red] at (O3) [circle through=(O5)] {} ;
    \node[green,fill=white] at ($(O1)!.15!(O3)$) {\ang{72}} ;
    \node[red,fill=white] at ($(O1)!.85!(O3)$) {\ang{216}} ;
    \node at ($(O2)!.2!(O6)$) {\ang{36}} ;
    \node at ($(O4)!.2!(O6)$) {\ang{36}} ;
    \end{scope}
}}}
\begin{document}
\begin{tikzpicture}
\pgfmathsetlengthmacro{\Len}{5cm};
\pgfmathsetlengthmacro{\Phi}{1.6180339};
\pgfmathsetlengthmacro{\PhiLen}{\Len * \Phi }; 
\pgfmathsetlengthmacro{\OverPhiLen}{\Len / \Phi };
\pgfmathsetlengthmacro{\BigAngLen}{ \Len / \Phi}; 
\pgfmathsetlengthmacro{\SmallAngLen}{ \BigAngLen / \Phi};
\foreach \s in {0,72,144,216,288} {%
\draw (1,0) pic [rotate=\s] {dart};
}

\end{tikzpicture}
\end{document}