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\module{Using congruent triangles}{01}{AP10}{1 period}

\prereq{Vocabulary and notions about triangles and angles.}

\object{\begin{itemize}
        \item Use congruent triangles to solve problems.
	\end{itemize}}

\mater{\begin{itemize}
	\item Exercises.
	\end{itemize}}

\modpart{Exercises}{Whole period}

Working by pairs, students have to solve some exercises and hand out a paper to the teacher at the end of the period.

\pagebreak


\moddocdis{Using congruent triangles}{01}{AP10}{Exercises}



\sectionblack{Exercise 1}
Let $ABC$ be a scalene triangle, with $CBFG$ and $ABDE$ two squares built outside of it
\begin{enumerate}
	\item Decompose the angles to prove that $\angle CBD=\angle ABF$.
	\item Deduce that $\triangle BCD$ and $\triangle ABF$ are congruent.
	\item Deduce that $AF=DC$.
\end{enumerate}


\sectionblack{Exercise 2}
Let $d_1$ and $d_2$ be two parallel lines, with $A$ a point on $d_1$ and $B$ a point on $d_2$. Let $K$ be the
midpoint of the segment $AB$. Through $K$, draw a line $\Dr$ that intersects $d_1$ in $M$ and $d_2$ in $N$.
\begin{enumerate}
	\item Prove that the triangles $KAM$ and $KBN$ are congruent.
	\item Deduce the position of $K$ on $MN$.
\end{enumerate}


\sectionblack{Exercise 3}
Let $ABC$ be a scalene triangle, with $I$ the midpoint of $BC$. Let $B'$ and $C'$ be the orthogonal projections of $B$
and $C$ onto the median $AI$ of the triangle.
\begin{enumerate}
\item Prove that the triangles $IBB'$ and $ICC'$ are congruent.
\item Deduce from the previous question the nature of $BBCC'$.
\end{enumerate}


\sectionblack{Exercise 4}
Let $ABCD$ be a square with center $O$ and $M$ a point on
the side $AB$. Let $P$ be the intersection of the side $AD$ and the line
perpendicular to $CM$ through $B$. 
\begin{enumerate}
\item Prove that $\angle DCM=\angle BMC$ and deduce that $\angle BCM=\angle ABP$.
\item Deduce that the triangles $MCB$ and $PBA$ are congruent and that $MB=AP$.
\item Prove that the triangles $OMB$ and $OPA$ are congruent.
\item Deduce that $\triangle POM$ is right-angled and isosceles.
\end{enumerate} 

%%%%%%%%%%%%%%Exos supp%%%%%%%%%%%%%%%
% \sectionblack{Exercise 3}
% Let $\Co$ ~and $\Co'$ two circle with respective centre $O$ and $O'$, with different radius, sécants en $A$ and $B$.
%$F$
% and $G$ are the les points diamétralement opposés à $A$ respectivement sur $\Co$ ~et $\Co'$, et $D$ et $E$ les points
% diamétralement opposés à $B$ respectivement sur $\Co$ ~et $\Co'$.
% \begin{enumerate}
% \item \begin{enumerate}
%       \item Prove that the triangles $AOO'$ and $BOO'$ are congruents.
%       \item Deduce three equalites.
%       \end{enumerate}
% \item \begin{enumerate}
%       \item Prove that the triangles $AFG$ and $BDE$ are congruents.
%       \item Deduce three length equalites.
%       \end{enumerate}
% \item \begin{enumerate}
%       \item Prove that the triangles $EOO'$ and $GOO'$ are congruents.
%       \item Deduce three equalites.
%       \end{enumerate}
% \item \begin{enumerate}
%       \item Prove that the triangles $DOO'$ and $FOO'$ are congruents.
%       \item Deduce three equalites.
%       \end{enumerate}
% \item Let $R$ and $P$ be the centre of gravity of triangles $AFG$ et $BDE$.\\
%       What could you say about triangles $ROO'$ and $POO'$ ? explain.
% \end{enumerate}

%bof car on ne montre pas grand chose, si ce n'est d'autre triangle isom
% \subsection*{Exercise 2}
% Let $ABCD$ be a diamond (or lozenge).
% \begin{enumerate}
% \item Draw the two points $E$ and $F$ such that
% $\vect{AE}=\frac23\vect{AD}$ and $\vect{AF}=\frac23\vect{AB}$.
% \item Prove that the triangles $ADF$ and $ABE$ are congruent.
% \item Deduce that the angles $\widehat{CDF}$ and $\widehat{CBE}$
% are equal.
% \item Deduce finally that $CBE$ and $CDF$ are congruent. Show the
% correspondance between the vertices.
% \end{enumerate}

\end{document}
