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\usepackage{amssymb, amsmath, amsthm}

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% This ends the preamble.  We now proceed to the document itself.

\begin{document}


   % title

   \title[short title]{long title}
   \author{}
   % Note that the short title for running heads goes in square
   % brackets.  This is optional.  The long title goes in curly
   % braces.  In the long title, line breaks are indicated by \\.


%Here are two sets of commands for including images.
  
%\begin{center}
%\begin{figure}[h]
%\includegraphics[scale=0.4]{FinalExam-prob10.pdf}
%\end{figure}
%\end{center}

%\begin{figure}[ht] 
%\scalebox{0.5}{\input{Pen.eps_t}}
%\caption{A pox on both your houses!}
%\end{figure}

   

   % today's date, or fill in whatever date you prefer
   \date{\today}

\framebox{
\raisebox{0in}[.5in][.2in]{
\textbf{MA 314 Homework 18: Putting a $K$ in your Gauss-Bonnet.} 
}
}

\section{Mortar}
\begin{enumerate}\spacing

\item Have a list of sources and an outline for your blog entry ready for Friday.

\item Read the essay on the Hyperbolic Pythagorean Theorem (written by yours truly) on the course blog. You will have to log into the blog website in order to see it. Come prepared to discuss a list of 3 aspects you like and 3 ways it could be improved. (Also turn in written versions of your comments)

\end{enumerate}

\section{Bricks}
\begin{enumerate}\spacing
\item Show that given $\alpha, \beta \in (0, \pi/2)$ such that $\alpha + \beta < \pi/2$, there exists a right triangle in $\H^2$ with angles $\alpha$ and $\beta$. 
(Hint: Take one side of the triangle to lie on the positive $x$-axis in the unit disc model. Take the other side to have angle $\alpha$ from the positive $x$-axis. Consider circles in $\R^2$ with center $(x_0,0)$ and radius $r>0$. Use the intermediate value theorem.)

\item Use the previous result to show that in $\H^2$, for any $n \geq 5$, there exists a regular $n$-gon in $\H^2$ having each internal angle equal to $2\pi/n$.  

\item Show how to give the torus the structure of a hyperbolic cone surface.

\item Do Problem 5.16 from Bonahon \textbf{and} explain its relevance for determining what kinds of geometric structures can be supported by a given surface.
\end{enumerate}
\end{document}