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\begin{document}

\begin{center}

\Huge{2 Dimensions Postulates}\\
\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

\begin{customPost}{1}
Any two points determine a unique line.
\end{customPost}

\begin{customPost}{2}
Given any two distinct points on a line, there is a one-to-one correspondence, called a \emph{ruler} between all points on the line with the real numbers that sends one of the two given points to 0, and the other to some number greater than 0.  The number $p$ assigned to a point $P$ by the ruler is called the \emph{coordinate}.
\end{customPost}

\begin{customPost}{3}
Every line $l$ determines a decompositions of the plane into 3 distinct sets: $H_0$, $H_1$, and $l$ where: 
\begin{packed_itemize}
\item Every pair of points in one of the $H_i$ are on the same side of $l$, and 
\item Every pair of points where one is in $H_0$ \& the other is in $H_!$ are on opposite sides of $l$.\end{packed_itemize} 
\end{customPost}

\begin{customPost}{4}
Let $\overrightarrow{OA}$ be given.  Then there is a 1-1 correspondence, called a \emph{protractor}, between all rays $\overrightarrow{OB}$ with the real numbers in the interval $(-\pi,\pi]$ that sends $\overrightarrow{OA}$ to $0$ and $\overrightarrow{OB}$ to some number $x\in (-\pi,\pi]$. The number $p$ assigned to a ray $\overrightarrow{OP}$ by the protractor is called its \emph{coordinate}. Furthermore:
\begin{packed_itemize}
\item $\overrightarrow{OB}$ has coordinate $0$ if and only if $\angle AOB$ is degenerate.
\item $\overrightarrow{OB}$ has coordinate $\pi$ if and only if $A-O-B$, that is if $\angle AOB$ is straight.
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\begin{customPost}{5}
Let $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ be distinct rays.  Then $\mu(\angle AOB)+\mu(\angle BOC)=\mu(\angle AOC)$ if and only if $B$ is in the interior of $\angle AOC$.
\end{customPost}

\begin{customPost}{6}
Given a line $l$, there is a unique bijective function $\phi$ from the plane to the plane called a \emph{fold with crease $l$} or a \emph{reflection with mirror $l$}, such that:
\begin{packed_enum}
\item The function $\phi$ leaves $l$ unchanged
\item Let $H_0$ and $H_1$ be the half planes determined by $l$. \\Then $A\in H_i$, implies $\phi(A) \in H_{1-i}$  for $i=0,1$.

\item If $\overline{AB}\subset \overline{H_i}$, then $\phi(A)\phi(B)=AB$
\item If $\angle ABC \subset \overline{H_1}$, then $\mu(\angle \phi(A)\phi(B)\phi(C)=\mu(\angle ABC)$
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\end{customPost}

\begin{customPost}{7}
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
\end{customPost}

\begin{customThm}{Playfaire's Theorem}
Given a line $l$, and a point $A\not \in l$, there is a unique line $m$ through $A$ and parallel to $l$.
\end{customThm}









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