Asymptote part
The first thing you need for that Asymptote code is a module called olympian, that can be downloaded here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=519&t=165767 This adds support for anglemark and more.
It is the file attached to the first post in that thread. Save it as olympian.asy in the same folder as your .tex file. To use the module, add import olympian; to the Asymptote code.
Next, you want to draw the arcs, so instead of having the anglemark within the labels, it seemed necessary to have it in a draw command, so
draw(anglemark(E,D,C));
draw(anglemark(D,E,A));
label("$\ \ \ \ \ \ \ 30^\circ$",D,SE);;
label("$80^\circ \ \ \ $",E,W);
E is the point (200,85).
The default size of the arc is quite small, it can be increased by adding
markscalefactor=3;
Adjust the value to your needs.
I don't know what is the best to label angles, but I've added a suggestion:
label("\small$30^\circ$",shift(35*dir(-7))*D);
This label is placed at D, but shifted 35 units in the direction -7 degrees. The \small command makes the font a little smaller. Perhaps not necessary, but it was a bit cramped in the top left corner.
Complete code:
\documentclass{standalone}
\usepackage[inline]{asymptote}
\begin{document}
\begin{asy}
import olympiad;
size(200);
pair A,B,C,D,E;
A=(0,0);
B=(200,0);
C=(200,150);
D=(0,150);
E=(200,85);
draw(A--B--C--D--cycle);
label("A",A,(-1,0));
label("D",B,(0,-1));
label("C",C,(1,0));
label("B",D,(0,1));
label("E",E,(1,0));
dot(E);
draw(A--E--D);
markscalefactor=3;
draw(anglemark(E,D,C));
draw(anglemark(D,E,A));
label("\small$30^\circ$",shift(35dir(-7))D);
label("\small$80^\circ$",shift(35dir(180))E);
\end{asy}
\end{document}
Compilation
When compiling the above with pdflatex, a file called <filename>-1.asy is generated. To generate the image, this has to be compiled with the Asymptote program, so run
asy <filename>-1.asy
from a command line/terminal. Compile your .tex file again with pdflatex and the image should be included in your document.
You could also open the .asy in TeXshop file and compile it with the Asymptote engine, mentioned by Alan Munn in a comment.
Finally, it should be possible to do this compilation automatically with latexmk, but I wasn't successful in my attempts (I probably did something wrong).
Output of above code

asyon the generated file? – Torbjørn T. Aug 19 '13 at 21:01asy"? My brain is not doing well today, sorry about that! – Ahaan S. Rungta Aug 19 '13 at 21:02filename-1.asy. You have to compile this with the Asymptote program, by issuingasy filename-1.asyin a terminal, which generates the figure, and then runpdflatexagain, to place it in the document. – Torbjørn T. Aug 19 '13 at 21:07.asyfile in the same folder as the.texfile. Now, when I open it, it just shows the Asy code with a few extra stuffages. From here, what do you mean by "issuingasy filename-1.asyin the terminal? – Ahaan S. Rungta Aug 19 '13 at 21:15