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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1316523248130--
%%--Sur les angles orientés--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Dans chacun des cas suivants, une mesure de l'angle orient\'{e} $\left(
\overrightarrow{u},\overrightarrow{v}\right)  $ est fix\'{e}e. Donner la
mesure principale de chacun des angles orient\'{e}s indiqu\'{e}s.

\begin{enumerate}
\item $\left(  \overrightarrow{u},\overrightarrow{v}\right)  =\dfrac{\pi}{2}.$%

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\begin{enumerate}
\item $\left(  -\overrightarrow{u},\overrightarrow{v}\right)  ;$

\item $\left(  -\overrightarrow{u},-\overrightarrow{v}\right)  ;$

\item $\left(  3\overrightarrow{u},2\overrightarrow{v}\right)  ;$

\item $\left(  \overrightarrow{v},\overrightarrow{u}\right)  .$
\end{enumerate}%

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\item $\left(  \overrightarrow{u},\overrightarrow{v}\right)  =-\dfrac{\pi}{6}.
$%

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\begin{enumerate}
\item $\left(  \overrightarrow{u},-\overrightarrow{v}\right)  ;$

\item $\left(  -\overrightarrow{u},\overrightarrow{v}\right)  ;$

\item $\left(  -\overrightarrow{v},\overrightarrow{u}\right)  ;$

\item $\left(  -\overrightarrow{v},-\overrightarrow{u}\right)  .$
\end{enumerate}%

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\item Construire une ligne bris\'{e}e ABCDE telle que:%
\begin{align*}
AB  & =4%
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\quad;\quad BC=3%
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\quad;\quad CD=2%
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\quad;\quad DE=2%
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;\\
\left(  \overrightarrow{BA},\overrightarrow{BC}\right)   & =-\dfrac{5\pi}{6}%
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\quad;\quad\left(  \overrightarrow{CB},\overrightarrow{CD}\right)
=-\dfrac{\pi}{2}%
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\quad;\quad\left(  \overrightarrow{DC},\overrightarrow{DE}\right)  =\dfrac
{\pi}{3}%
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.
\end{align*}


\item D\'{e}montr\'{e} que $\overrightarrow{AB}$ et $\overrightarrow{DE}$ sont colin\'{e}aires.
\end{enumerate}
\end{exercise}

\begin{exercise}
Soit $AOI$ un triangle \'{e}quilat\'{e}ral avec $\left(  \overrightarrow
{A0},\overrightarrow{AI}\right)  =\pi/3.$ Les points $J$ et $B$ sont tels que
les triangles $OIJ$ et $IBA$ sont rectangles isoc\`{e}les avec $\left(
\overrightarrow{OI},\overrightarrow{OJ}\right)  =\left(  \overrightarrow
{IB},\overrightarrow{IA}\right)  =\pi/2.$

\begin{enumerate}
\item Faire une figure faisant appara\^{\i}tre toutes les donn\'{e}es;

\item
\begin{enumerate}
\item Calculer une mesure des angles g\'{e}om\'{e}triques $\widehat{JAO}$ et
$\widehat{IAB};$

\item En d\'{e}duire une mesure des angles orient\'{e}s $\left(
\overrightarrow{AJ},\overrightarrow{AO}\right)  $ et $\left(  \overrightarrow
{AI},\overrightarrow{AB}\right)  .$
\end{enumerate}

\item En d\'{e}duire une mesure de l'angle orient\'{e} $\left(
\overrightarrow{AJ},\overrightarrow{AB}\right)  .$ Conclure.
\end{enumerate}
\end{exercise}


\end{document}