\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}{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{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=0.6in,right=0.7in,top=0.7in,bottom=0.9in} \pagestyle{fancy} \lhead{\color{blue} \LARGE Lecture Notes} \lfoot{\small \copyright \; Hidegkuti, Powell, 2009} \rfoot{\small Last revised: June 14, 2013} \chead{\huge Angles in a Triangle} \rhead{page \thepage} \cfoot{} \textwidth 7.4in \textheight 9.5in \setlength{\headheight}{25pt} \setlength{\parindent}{0in} \begin{document} In mathematics, we will be dealing with different types of true statements. \ \ Some examples for these are definitions, axioms and theorems. \ What are these? \ \bigskip A \textbf{definition} is a type of statement in which we agree how we will refer to things. \ It is true in a sense because it just sets an agreement about labeling things. \ \bigskip An \textbf{axiom} is a statement that we accept as true, without requiring proof of it. \ It usually agrees with our natural instincts and they "feel true". One example is the statement: "Two points uniquely determine a straight line". \bigskip A \textbf{theorem} is a statement that we prove to be true. But what does it mean to prove something? It means to derive it from the axioms. \ Mathematicians set down a set of basic 'truths', the axioms. Everything we prove, we derive them from the axioms. \ The following proof is a presentation of this. We will use two axioms to prove a theorem. Here is all we need:\bigskip \textbf{Definition:}\qquad The angle shown on the picture below is called the straight angle and it measures\textit{\ }$180^{\circ }$\textit{.\FRAME{% dtbpF}{3.0943in}{0.8899in}{0pt}{}{}{pic1.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.0943in;height 0.8899in;depth 0pt;original-width 0.0969in;original-height 0.0251in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../Desktop/pic1.bmp';file-properties "XNPEU";}}}Axiom 1:\qquad If two straight lines intersect each other, then the opposite angles formed have equal measures as the picture below shows.\FRAME{dtbpF}{2.047in}{% 1.9804in}{0pt}{}{}{pic2.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.047in;height 1.9804in;depth 0pt;original-width 0.0761in;original-height 0.0822in;cropleft "0.0769";croptop "0.9568";cropright "0.9231";cropbottom "0.0432";filename '../../../../../Desktop/pic2.bmp';file-properties "XNPEU";}}These angles are called vertical angles.\bigskip Axiom 2:\qquad If $e$\ and $e^{\prime }$\ are parallel lines, and $l$\ is a line intersecting these lines, then the two angles marked on the picture below have equal measures. \ We say that line $l$\ is a transversal for the lines $e$\ and $e^{\prime }$.\FRAME{dtbpF}{3.6547in}{2.1266in}{0pt}{}{}{% pic3b.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.6547in;height 2.1266in;depth 0pt;original-width 0.173in;original-height 0.0891in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../Desktop/pic3b.bmp';file-properties "XNPEU";}} \pagebreak We are now ready to state and prove our first theorem. \bigskip Theorem: \ If \ $\alpha ,$ \ $\beta ,$ \ and \ $\gamma $ \ are the three angles of a triangle, then $\alpha +\beta +\gamma =180^{\circ }$.\bigskip \begin{proof} Let $ABC$ be any triangle. Let us denote the angles by $\alpha ,$ $\beta ,$ and $\gamma ,$ and the sides by $a,$ $b,$ and $c$ as shown on the picture below.\FRAME{dtbpF}{2.2079in}{1.4278in}{0pt}{}{}{insert4.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.2079in;height 1.4278in;depth 0pt;original-width 2.1793in;original-height 1.4001in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert4.bmp';file-properties "XNPEU";}}Let us draw the lines $a$, $b$, and $c$ longer, beyond the triangle. \ As often times in geometry, proofs are based on a smartly drawn, single line. In this case, the 'magical line' that will give us the proof, is a line, we will call it $c^{\prime },$ that is parallel to $c$ and passes throught the point $C$. There are three new angles formed, we will label them as $\angle 1,$ $\angle 2,$ and $\angle 3,$ as shown on the picture below.\bigskip \FRAME{dtbpF}{2.6247in}{2.188in}{0pt}{}{}{insert5.bmp}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.6247in;height 2.188in;depth 0pt;original-width 2.1793in;original-height 1.8135in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert5.bmp';file-properties "XNPEU";}}Because of Axiom 1, $\gamma =\angle 2$. Those two angles are vertical as the lines $a$ and $b$ intersect. Because $a$ is a transversal for the parallel lines $c$ and $c^{\prime }$, $\angle 1=\beta $. Because $b$ is a transversal for the parallel lines $c$ and $c^{\prime }$, $\angle 3=\alpha $. \ We can observe that $\angle 1+\angle 2+\angle 3=180^{\circ },$ since they are forming a straight angle together. So we have:% \begin{eqnarray*} \underset{\beta }{\underbrace{\angle 1}}+\underset{\gamma }{\underbrace{% \angle 2}}+\underset{\alpha }{\underbrace{\angle 3}} &=&180^{\circ } \\ \alpha +\beta +\gamma &=&180^{\circ } \end{eqnarray*}% This concludes our proof. \end{proof} \end{document}