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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Angles In the Plane">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Feb 12 15:48:22 1997}
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\section{Angles in the Plane}

The \label{Standard position}\textbf{standard position} of an angle occurs
when the vertex is placed at the origin of a coordinate system and its
initial side is the positive $x$-axis. A \label{Positive angle}\textbf{%
positive} angle is obtained by rotating the initial side counterclockwise
until it coincides with the terminal side. A \label{Negative angle}\textbf{%
negative} angle is obtained by clockwise rotation.

\begin{itemize}
\item  Different angles can have the same terminal side. For example, the
angles $\frac{3}{4}\pi $, $-\frac{5}{4}\pi $, and $\frac{11}{4}\pi $ have
the same terminal sides because they differ by integer multiples of $2\pi $.
Such angles are called \label{Coterminal angles}\textbf{coterminal}.

\item  An angle whose terminal side coincides with one of the axes is called
a \label{Quadrantal angle}\textbf{quadrantal} angle. Some quadrantal angles
are $0\unit{\UNICODE{0xb0}}$, $90\unit{\UNICODE{0xb0}}$, $180\unit{%
\UNICODE{0xb0}}$, $270\unit{\UNICODE{0xb0}}$, $360\unit{\UNICODE{0xb0}}$, $%
-90\unit{\UNICODE{0xb0}}$, and $-180\unit{\UNICODE{0xb0}}$ (or $0$, $\frac{%
\pi }{2}$, $\pi $, $\frac{3\pi }{2}$, $2\pi $, $-\frac{\pi }{2}$, and $-\pi $%
).
\end{itemize}

\subsubsection{Angles In a Plane\label{Angles in a plane}}

Let $P_{1}(x_{1},y_{1})$ and $P_{2}(x_{2},y_{2})$ be any two points in a
plane, and let $\theta $ be the angle measured counter-clockwise from the
line $0X$ to the line $P_{1}P_{2}$.

\begin{itemize}
\item  \label{Slope of line}The \textbf{slope} of the line $P_{1}P_{2}$ is
given by 
\[
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\tan \theta 
\]

\item  \label{Angle between lines}The \textbf{angle }$\theta $\textbf{\
between lines} of slopes $m_{1}$ and $m_{2}$, respectively, is given by 
\[
\tan \theta =\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}
\]

\item  Two lines of slopes $m_{1}$ and $m_{2}$ are \label%
{Perpendicular lines}\textbf{perpendicular} if $m_{2}=-\dfrac{1}{m_{1}}$, 
\label{Parallel lines}\textbf{parallel} if $m_{1}=m_{2}$.
\end{itemize}

\subsection{\ Related Reference Topics}

{\small \hyperref{Table of Geometric Formulas For Length, Area, and Volume}{%
}{}{TAB_geom_formulas.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Angular Measure}{}{}{../../Help/DMIndex.tex#Angular measure%
}}

{\small \hyperref{Conic Sections}{}{}{../../Help/DMIndex.tex#Conic sections}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Polygons}{}{}{../../Help/DMIndex.tex#Polygons}}

{\small \hyperref{Surfaces of Revolution}{}{}{%
../../Help/DMIndex.tex#Surfaces of revolution}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}\hfill

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