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% This is a template for students in MATH 3000 at FSU to formally write-up results presented in class.
% All of this stuff with '%' in front is a comment and ignored by the compiler.
%
% The lines before the "\begin{document}" line is called the preamble.
% This is where you load particular packages you need.
% Until you are more experienced, or the program says you are missing packages, it is safe to ignore most of the preamble.
%
%----------------------------------
\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}% Change the margins here if you wish.
\setlength{\parindent}{0pt} % This is the set the indent length for new paragraphs, change if you want.
\setlength{\parskip}{5pt} % This sets the distance between paragraphs, which will be used anytime you have a blank line in your LaTeX code.
\pagenumbering{gobble}% This means the page will not be numbered. You can comment it out if you like page numbers.
%------------------------------------
% These packages allow the most of the common "mathly things"
\usepackage{amsmath,amsthm,amssymb}
% This package allows you to add images.
\usepackage{graphicx}
\usepackage{float}
% These are theorem environments. This should cover everything you need, and you should be able to tell what environment goes with what type of result, but please let me know if I've missed anything.
\newtheorem{euclidtheorem}{Proposition}[section]
\newtheorem{classtheorem}{Theorem}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{challenge}[theorem]{Challenge}
\newtheorem{question}[theorem]{Question}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{theorempiece}{Theorem}[theorem]
\newtheorem{classtheorempiece}{Theorem}[classtheorem]
\newtheorem{challengepiece}{Challenge}[theorem]
\newtheorem{questionpiece}{Question}[theorem]
\newtheorem{problempiece}{Problem}[theorem]
%These help to format the names of the results the way we are in class and in notes.
\renewcommand*{\theeuclidtheorem}{\Roman{section}.\arabic{theorem}}
\renewcommand*{\thetheorem}{\arabic{section}.\arabic{theorem}}
\renewcommand*{\theclasstheorem}{\Alph{theorem}}
\renewcommand*{\thechallenge}{\arabic{section}.\arabic{theorem}}
\renewcommand*{\thequestion}{\arabic{section}.\arabic{theorem}}
\renewcommand*{\theproblem}{\arabic{section}.\arabic{theorem}}
\renewcommand*{\thetheorempiece}{\arabic{section}.\arabic{theorem}.\alph{theorempiece}}
\renewcommand*{\thechallengepiece}{\arabic{section}.\arabic{theorem}.\alph{challengepiece}}
\renewcommand*{\thequestionpiece}{\arabic{section}.\arabic{theorem}.\alph{questionpiece}}
\renewcommand*{\theproblempiece}{\arabic{section}.\arabic{theorem}.\alph{problempiece}}
% Should you need any additional packages, you can load them here. If you've looked up something (like on DeTeXify), it should specify if you need a special package. Just copy and paste what is below, and put the package name in the { }.
\usepackage{wasysym} %this lets me make smiley faces :-)
% Put the name of your paper here. It does not need to be a fancy name, but should tell the reader what is contained in the paper.
\title{Exterior Angle Theorem}
% You are the author, put your name here.
\author{Sarah Wright}
% You can change the date to be something other than the current date if you want.
\date{\today}
\begin{document}
\maketitle
Write a (very brief) introduction here. You should include the name of the original presenter of the result, and (if appropriate) the name of the person who posed the question or conjecture originally. Also include the names of any other helpful persons to your write-up, the presentation, or the solution itself.
Include some narrative that will connect this result to others or give some of the background on the questions or conjectures the result answers. (This will be easier in some cases than others; don't worry too much about being eloquent here \dots focus on the art of proof-writing first.)
% These set the counters for the section (of our notes; so section 1 is rhombuses, section 2 is kites, etc.) and theorem (or problem, challenge, etc.) If you are only presenting a piece of a theorem that is not the first piece, you'll need to uncomment the last counter and adjust it appropriately.
\setcounter{section}{1}
\setcounter{theorem}{15}
%\setcounter{theorempiece}{2}
% Use the appropriate environment to get the name that you want.
\begin{euclidtheorem}
The exterior angle of a triangle is greater than either of the interior and opposite angles.
\end{euclidtheorem}
% Every environment should have a \begin{} and \end{} where the inside of {} matches.
\begin{proof} % This environment prints "Proof." at the beginning and the box at the end.
Let $ABC$ be a triangle, and extend segment $BC$ to point $D$ by Euclid's Postulate 2. We wish to show that exterior angle $ACD$ is greater than each of the interior angles $CBA$ and $BAC$.
% Notice that stuff inside $ $ looks different in the pdf. This is math mode, and you should use it for the names of mathematical objects, and you HAVE to use it for almost every mathematical symbol.
Construct $E$ to be the midpoint of segment $AC$, by Euclid I.10, and construct the segment $BE$ by Euclid's Postulate 1 and extend $BE$ beyond point $E$ by Euclid's Postulate 2. Using Euclid I.3, construct the point $F$ on the extended segment $BE$ such that $BE$ is congruent to $EF$. Construct the segment $CF$ by Euclid's Postulate 1.
\setcounter{figure}{14}
\begin{figure}[H]% This environment labels and generates a figure separate from the text. I used the [H] to place the figure exactly where I wanted it. If this isn't important to the reading and understanding of your proof, you can let LaTeX choose where to put it by leaving out the [] altogether. You can use other tags to control the placement of a figure, [t] for the top of a page and [b] for the bottom are convenient sometimes.
\centering
\includegraphics[width = .4\textwidth]{I_16_picture}
% This is how we include pictures in a LaTeX document. In Overleaf, you need to add this file to the project. Click on "Project" with the squares in the menu at the top. Choose "Files..." and probably upload from your computer. (If you have installed a LaTeX editor on your own machine, then make sure the file you want to include is in the same folder as your .tex document... or look up how to call other folders.)
% Once the file is part of the project, type the name of the file inside the {}. You typically do not need the file type .pdf, .jpg, etc. but nothing bad happens when you include it.
% The [] can be left empty or used to enter a variety of different instructions, most commonly adjustments to the size of the image are here. You can use inches or centimeters or points, LaTeX knows many measurement systems. You can also specify relative lengths. \textwidth is the width of the text in this document, and putting .65 in front shrinks it to 85% of the text width. You can also give a scale rather than a width or height: [scale = .5] will shrink the entire image to half the size of the original.
\label{fig:1.15}
% This is a label we can use to reference this figure... more helpful if you have lots of figures in your proof.
\caption{$\overline{AE} \cong \overline{EC}$, $\overline{BE} \cong \overline{EF}$, $\overline{BH} \cong \overline{HC}$, and $\overline{AH} \cong \overline{HI}$.}
% I like to caption the pictures in my geometry proofs with the congruences that are "constructed in" to the figure. I basically include the "givens" and allow the reader to verify the conclusions by reading the proof and looking at the diagram together. You can of course make different choices.
\end{figure}
Segments $AE$ and $EC$ are congruent by construction; as are segments $BE$ and $EF$. Since angles $AEB$ and $CEF$ are vertical angles, they are also congruent by Euclid I.15.
Thus, by Euclid I.4 (SAS congruence) triangles $BAE$ and $CEF$ are congruent and angle $EAB$ is congruent to angle $ECF$.
Angle $ECD$ is greater than angle $ECF$ by Euclid's Common Notion 5. Therefore, angle $EAB$ is greater than angle $ECD$.
This shows that the exterior angle, namely $ACD$, is greater than one of the interior opposite angles, namely $CAB$.
To show that the exterior angle is also greater than the remaining opposite interior angle, $ABC$, similarly extend segment $AC$ to $G$, bisect segment $BC$ at $H$, construct the segment $AH$, and extend to $I$ such that segments $AH$ and $HI$ are congruent. The analogous argument shows that triangles $ABH$ and $ICH$ are congruent, making angles $HBA$ and $HCI$ congruent. But, angles $ACD$ and $HCG$ are congruent, and angle $HCG$ is greater than $HCI$. This shows that the exterior angle $ACD$ is greater than the opposite interior angle $ABC$ as desired. \end{proof}
If there is any narrative that makes sense here, or closing remarks, feel free to include something. $\smiley$
\end{document}
```