\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.5in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture Notes} \chead{\color{black} \LARGE Standard Labeling} \rhead{\large } \cfoot{} \lfoot{\small \copyright $\;$ Hidegkuti, Powell, 2008} \textwidth 7.4in \textheight 9.5in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} Standard labeling is an agreement among mathematicians on how we label sides, vertices, and angles of triangles. \ Every triangle has three of the following three components: \begin{enumerate} \item \textbf{vertices}\newline Points are usually denoted by uppercase letters. \ In case of triangles, we often use $A$, $B$, and $C$. \item \textbf{angles}\newline Angles are usually denoted by lowercase Greek letters. In case of triangles, we often use $\alpha $ (alpha), $\beta $ \ (beta), and $\gamma $ (gamma). \item \textbf{sides}\newline Lines and line segments are usually denoted by lowercase letters. \ In case of triangles, we often use $a$, $b$, and $c$. \end{enumerate} In case of standard labeling, we automatically associate sides, vertices, and angles. \ A vertex is associated with the angle located at that vertex. These two are associated with the side opposite these. \ For example, angle $% \alpha $ is always assumed to be located at point $A,$ and side $a$ is always assumed to be the side oppsite to point $A$ and angle $\alpha $. \ Point $B,$ angle $\beta ,$ and side $b$ are similarly grouped. \ The figure below summerizes standard labeling. \ \FRAME{dtbpF}{3.1669in}{2.0479in}{0pt}{% }{}{standard.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.1669in;height 2.0479in;depth 0pt;original-width 2.2935in;original-height 1.4728in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'standard.bmp';file-properties "XNPEU";}}Unless otherwise indicated, we should always assume standard labeling when presented with data, using these letters. \vspace{2.7in}{\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}