\documentclass [12pt] {article}

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\newcustomtheorem{customPrac}{Practice}


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\begin{document}

\begin{center}

\Huge{1 Dimensional Geometry}\\
\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}{L-1}
Given any two distinct points, there is a 1-1 correspondence, called a \emph{ruler}, between all points with the real numbers that sends one of the two given to $0$ and the other to some number $x>0$.  The number $p$ assigned to a point $P$ by the ruler is called its \emph{coordinate}.
\end{customPost}
\vspace{.5in}

\begin{customDef}{2.1}
If $A$ and $B$ are arbitrary points with coordinates $a$ and $b$, then the \emph{distance} $AB$, from $A$ and $B$ is defined to be $AB=\vert a-b\vert$.
\end{customDef}

\begin{customDef}{2.2}
Let $A$, $B$, and $C$ be distince points with coordinates $a$, $b$, and $c$ respecitively.  Then $B$ is \emph{between} $A$ and $C$ (denoted $A-B-C$) provided that either $a<b<c$ or $c<b<a$.
\end{customDef}
\vspace{.5in}


\begin{customThm}{2.1}
If $B$ is between $A$ and $C$, then $AB+BC=AC$.
\end{customThm}


\vspace{3in}

\newpage
\begin{customDef}{2.3}
Given two distince points $A$ and $B$ we define the \emph{segment} (denoted $\overline{AB}$) to be the set of points $C$ so that either $C=A$, $C=B$, or $A-C-B$, i.e. $$\overline{AB}=\{C:C=A, C=B, \text{ or } A-C-B\}.$$
The \emph{length} of a segment is the distance between its endpoints.
\end{customDef}


\begin{customDef}{2.4}
Given two points $A$ and $B$ we define the \emph{ray} (denoted $\overrightarrow{AB}$) to be the set of points $C$ so that either $C=A$, $C=B$, $A-C-B$, or $A-B-C$, i.e. $$\overrightarrow{AB}=\{C: C=A, C=B, A-C-B, \text{ or } A-B-C \}.$$
\end{customDef}

\begin{customPrac}{2.4}
Suppose we define ``points" as non-vertical lines through the origin in $\mathbb{R}^2$.
\end{customPrac}

\begin{enumerate}
\item Verify that Postulate L-1 is satisfied.
\vspace{1in}
\item Sketch the segment between $0$ and $1$.
\vspace{2in}
\end{enumerate}


\begin{customPost}{L-2}
Let O be a point.  There is a bijective function $\phi$ from the line to the line called a \emph{fold with crease O} such that:
\begin{packed_enum}
\item $\phi(O)=O$.
\item Let $\overrightarrow{OA_0}$ and $\overrightarrow{OA_1}$ be the two rays determined by O.  Then B$\in \overrightarrow{OA_i}$ implies $\phi(B)\in \overrightarrow{OA_{1-i}}$.
\item If segment $\overline{BC}\subset\overrightarrow{OA_i}$, then $BC=\phi(B)\phi(C)$.
\end{packed_enum}
\end{customPost}

\begin{customDef}{2.6}
Let $A$ and $B$ be distinct points, and let $a$ and $b$ be their respective coordinates.  Let $M$ be the point with coordinate $\frac{a+b}{2}$.  Then $M$ is called the \emph{midpoint} of the segment $\overline{AB}$.
\end{customDef}


\end{document}




