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\begin{document}

\begin{center}

\Huge{2 Dimensional Euclid}\\
\normalsize{definitions \& theorems from Origametry by Daniel Heath.}

\end{center}

\noindent
While working in a group make sure you:
\begin{packed_itemize}
\item Expect to make mistakes but be sure to reflect/learn from them!
\item Are civil and are aware of your impact on others.
\item Assume and engage with the strongest argument while assuming best intent.
\end{packed_itemize}

\normalsize

\vspace{.3in}


\begin{enumerate}

\item Given a line $l$, on patty paper and a point $A$ not on the line, what steps can you take to fold a line through $A$ that is parallel to $l$?
\vspace{2.5in}

\item Begin with two parallel lines on a sheet of patty paper an fold any transversal over the parallel lines.  What can you say about the alternating interior angles?
\vspace{1.5in} 



\begin{customThm}{6.4}
If two distinct lines $l_1$ and $l_2$ are parallel, then there is a unique fold $\phi$ that takes $l_1$ to $l_2$.
\end{customThm}


\newpage
\item Begin with patty paper with two parallel lines $l$ and $m$ and a point $A$ not on either line.  Let $\phi_l$ be the fold with crease $l$ and $\phi_m$ be the fold with crease $m$.
\begin{enumerate}
\item Find $\phi_l(A)=A'$ and $\phi_m(A')=B$.
\vspace{.5in}
\item Fold the line $\overleftrightarrow{AB}$ and mark points $C=l\cap \overleftrightarrow{AB}$ and $D=m\cap \overleftrightarrow{AB}$.
\vspace{.5in}
\item Find a relationship between $AB$ and $CD$ that is true in general.
\vspace{3.5in}
\end{enumerate}


\begin{customThm}{6.7}
Let $l$ and $m$ be parallel lines, and $\phi_l$ and $\phi_m$ be the fold with crease $l$ and $m$ respectively.  Let $A$ be any point and $B=\phi_m\circ \phi_l$. Then $l \perp \overleftrightarrow{AB}$ and $AB=??$
\end{customThm}





\end{enumerate}





\end{document}




