Labelling Angle Issues

After some efforting on your code, following output can be obtained.

documentclass[margin=3mm]{standalone}
usepackage{tkz-euclide}
usetikzlibrary{calc}
%usetkzobj{all}
begin{document}
begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt},>=stealth]
coordinate (C) at (0,0) ;
coordinate (A) at (80:5);
coordinate (B) at (20:9) ; 
node [dot] at (C) {};
node [dot] at (A) {};
node [dot] at (B) {};
draw[-] (A)node[above]{A}--(C)node[left]{C};
draw[-] (C)--(B)node[right]{B};
draw[-] (A)--(B);
coordinate (Q) at($(C)!(A)!(B)$) ;
tkzMarkRightAngle(A,Q,C);
draw[purple!70!black] (A)--(Q);
draw[|<->|] ($(Q)!-4.9mm!90:(A)$)--node[fill=white] {10} ($(A)!3mm!90:(Q)$);
draw[|<->|] ($(C)!-7mm!90:(B)$)--node[fill=white,sloped] {$x$} ($(B)!7mm!90:(C)$);
tkzMarkAngle[size=0.75,draw = black, fill = white, opacity=1](Q,C,A)
tkzLabelAngle[pos=1,font=scriptsize](Q,C,A){$alpha$}
tkzMarkAngle[size=0.75,draw = black, fill = white, opacity=1](A,B,Q)
tkzLabelAngle[pos=1,font=scriptsize](A,B,Q){$beta$}
tkzMarkRightAngle(B,A,C);

end{tikzpicture}
end{document}

Using the code from Angle Label Issues, and commenting out the draw that uses the undefined coordinate P, I can reproduce the problem with TeX Live 2019 (on Overleaf)

I first thought it was just a matter of using tkzLabelAngle, but that gave initially the same, erroneous, result. Swapping the order of the coordinates, i.e. using (Q,B,A) instead of (A,B,Q) makes no difference either. In the new version of tkz-euclide, demonstrated in ferahfeza's answer, this works as expected, so if there was a bug, it has been fixed.

A workaround for the old version is to use a negative value for pos, in the options to tkzLabelAngle:

tkzMarkAngle[size=1cm,color=black](A,B,Q);  
tkzLabelAngle[pos=-1.2](Q,B,A){$beta$};  

documentclass{standalone}  
usepackage{tkz-euclide}  
usetkzobj{all}  
begin{document}  
begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}]  
path  
  (80:5) node [dot,label=above left:$A$]{} coordinate (A)  
  (20:9) node[dot, label=right:$B$]{} coordinate (B)  
  (0:0) node[dot,label=below left:$C$]{} coordinate(C);  

path coordinate (Q)at($(C)!(A)!(B)$) node at (Q)[]{} ;  

 draw[-]  (A)--(C);  
 draw[-] (C)--(B);  
 draw[-] (A)--(B);  
 draw[purple!70!black] (A)--(Q);  

tkzMarkRightAngle(A,Q,C);  

%draw[|<->|] ($(P)!-7mm!90:(B)$)--node[fill=white,sloped] {$x$} ($(B)!-7mm!-90:(P)$);  
draw[|<->|] ($(Q)!-4.9mm!90:(A)$)--node[fill=white] {$10$} ($(A)!-3mm!-90:(Q)$);  

tkzMarkAngle[size=0.75cm,color=black,label=a](B,C,A);  
tkzMarkAngle[size=1cm,color=black](A,B,Q);  
tkzLabelAngle[pos=-1.2](Q,B,A){b};  
tkzMarkRightAngle(B,A,C);  
end{tikzpicture}  
end{document}  

Using only pure tkz-euclide

documentclass{article} % or another class
usepackage{xcolor} % before tikz or tkz-euclide if necessary

usepackage{tkz-euclide} % no need to load TikZ

begin{document}
begin{tikzpicture}[scale=1.0]
    %define points A,B,C
    tkzDefPoint(0,0){C}
    tkzDefPoint(20:9){B}
    tkzDefPoint(80:5){A}
    %label point A,B,C
    tkzLabelPoints(B,C)
    tkzLabelPoints[above](A)
    %draw triangleABC
    tkzDrawPolygon[thick,fill=yellow!15](A,B,C)
    %get line orthogonal to base CB
    tkzDefPointsBy[projection=onto B--C](A){a}
    tkzDrawSegment[dashed, red](A,a)
    %marking right angles    
    tkzMarkRightAngle(A,a,C)    
    tkzMarkRightAngle(C,A,B)
    %drawing dimension 10
    tkzDrawSegment[style=red, dashed, dim={$10$,15pt,midway,font=scriptsize, rotate=90}](A,a) 
    %marking the angles
    tkzFillAngle[fill=blue!20, opacity=0.5](B,C,A)
    tkzLabelAngle[pos=1.25](B,C,A){$alpha$}
    tkzMarkAngle(B,C,A)
    tkzFillAngle[fill=red!20, opacity=0.5](A,B,C)
    tkzLabelAngle[pos=1.25](A,B,C){$beta$}
    tkzMarkAngle(A,B,C)
    
    
        tkzDrawPoints(A,B,C)

end{tikzpicture}
end{document}