\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}
\en
\usetikz

\begin{document}

\module{Congruent triangles}{01}{AP09}{1 period}
\nocouleur

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

\object{\begin{itemize}
        \item Discover the concept of congruent triangles.
	\item Apply out the three congruence cases to prove that some triangles are congruent.
	\end{itemize}}

\mater{\begin{itemize}
	\item Slideshow.
	\item Lesson.
	\item Exercises.
	\item Slideshow with the correction of the exercises.
	\end{itemize}}

\modpart{Lesson}{15 mins}

Using a slideshow, the teacher explains the concept of congruent triangles and the three congruence cases. Students
should have an active part to the lesson. At the end, a lesson sheet is handed out. 

\modpart{Exercises}{Remaining time}%{30 mins}

Working in groups, students have to work on a few exercises about congruent triangles.

% \modpart{Correction}{Remaining time}
% Using a slideshow, the teacher displays a correction so that each students could correct his mistake and get a template to answer such exercise.

\pagebreak


\moddocdis{Congruent triangles}{01}{AP09}{Lesson}

\begin{multicols}{2}

\defi{Equal sides}{Two triangles are congruent if they have three pairs of equal sides.}

\prop{Equal angles}{Two congruent triangles have three pairs of equal angles.}


\begin{center}
\newrgbcolor{qrqrqr}{0 0 0}
\psset{xunit=0.6cm,yunit=0.6cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.48,-0.82)(9.06,5.9)
\psline[linecolor=qrqrqr](-2.54,2.7)(1.56,5.08)
\psline[linecolor=qrqrqr](-0.54,3.97)(-0.44,3.81)
\psline[linecolor=qrqrqr](1.56,5.08)(2.8,2.18)
\psline[linecolor=qrqrqr](2.25,3.7)(2.08,3.63)
\psline[linecolor=qrqrqr](2.28,3.63)(2.11,3.56)
\psline[linecolor=qrqrqr](2.8,2.18)(-2.54,2.7)
\psline[linecolor=qrqrqr](0.19,2.34)(0.21,2.52)
\psline[linecolor=qrqrqr](0.12,2.35)(0.14,2.53)
\psline[linecolor=qrqrqr](0.05,2.36)(0.07,2.54)
\psline[linecolor=qrqrqr](2.68,0.54)(6.78,2.92)
\psline[linecolor=qrqrqr](4.68,1.81)(4.78,1.65)
\psline[linecolor=qrqrqr](6.78,2.92)(8.02,0.02)
\psline[linecolor=qrqrqr](7.47,1.54)(7.3,1.47)
\psline[linecolor=qrqrqr](7.5,1.47)(7.33,1.4)
\psline[linecolor=qrqrqr](8.02,0.02)(2.68,0.54)
\psline[linecolor=qrqrqr](5.41,0.18)(5.43,0.36)
\psline[linecolor=qrqrqr](5.34,0.19)(5.36,0.37)
\psline[linecolor=qrqrqr](5.27,0.2)(5.29,0.38)
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{-0.09707221915662737}{
0.5259487304013318}{0.6*cos(t)+-2.54|0.6*sin(t)+2.7}\lineto(-2.54,2.7)\closepath}
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{-0.09707221915662735}{
0.5259487304013317}{0.6*cos(t)+2.68|0.6*sin(t)+0.54}\lineto(2.68,0.54)\closepath}
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{-2.615643923188461}{
-1.1667371797845036}{0.6*cos(t)+6.78|0.6*sin(t)+2.92}\lineto(6.78,2.92)\closepath}
\parametricplot[linecolor=qrqrqr]{-2.615643923188461}{-1.1667371797845036}{0.6*cos(t)+6.78|0.6*sin(t)+2.92}
\parametricplot[linecolor=qrqrqr]{-2.615643923188461}{-1.1667371797845036}{0.5*cos(t)+6.78|0.5*sin(t)+2.92}
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{-2.615643923188461}{
-1.1667371797845039}{0.6*cos(t)+1.56|0.6*sin(t)+5.08}\lineto(1.56,5.08)\closepath}
\parametricplot[linecolor=qrqrqr]{-2.615643923188461}{-1.1667371797845039}{0.6*cos(t)+1.56|0.6*sin(t)+5.08}
\parametricplot[linecolor=qrqrqr]{-2.615643923188461}{-1.1667371797845039}{0.5*cos(t)+1.56|0.5*sin(t)+5.08}
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{1.9748554738052893}{
3.0445204344331662}{0.6*cos(t)+2.8|0.6*sin(t)+2.18}\lineto(2.8,2.18)\closepath}
\parametricplot[linecolor=qrqrqr]{1.9748554738052893}{3.0445204344331662}{0.6*cos(t)+2.8|0.6*sin(t)+2.18}
\parametricplot[linecolor=qrqrqr]{1.9748554738052893}{3.0445204344331662}{0.5*cos(t)+2.8|0.5*sin(t)+2.18}
\parametricplot[linecolor=qrqrqr]{1.9748554738052893}{3.0445204344331662}{0.4*cos(t)+2.8|0.4*sin(t)+2.18}
\pscustom[linecolor=qrqrqr,fillcolor=qrqrqr]{\parametricplot{1.9748554738052895}{
3.0445204344331662}{0.6*cos(t)+8.02|0.6*sin(t)+0.02}\lineto(8.02,0.02)\closepath}
\parametricplot[linecolor=qrqrqr]{1.9748554738052895}{3.0445204344331662}{0.6*cos(t)+8.02|0.6*sin(t)+0.02}
\parametricplot[linecolor=qrqrqr]{1.9748554738052895}{3.0445204344331662}{0.5*cos(t)+8.02|0.5*sin(t)+0.02}
\parametricplot[linecolor=qrqrqr]{1.9748554738052895}{3.0445204344331662}{0.4*cos(t)+8.02|0.4*sin(t)+0.02}
\psdots[linecolor=qrqrqr](-2.54,2.7)
\rput[bl](-3.1,2.56){\qrqrqr{$A$}}
\psdots[linecolor=qrqrqr](1.56,5.08)
\rput[bl](1.42,5.3){\qrqrqr{$B$}}
\psdots[linecolor=qrqrqr](2.8,2.18)
\rput[bl](2.98,1.88){\qrqrqr{$C$}}
\psdots[linecolor=qrqrqr](6.78,2.92)
\rput[bl](6.86,3.04){\qrqrqr{$E$}}
\psdots[linecolor=qrqrqr](2.68,0.54)
\rput[bl](2,0.38){\qrqrqr{$D$}}
\psdots[linecolor=qrqrqr](8.02,0.02)
\rput[bl](8.3,-0.16){\qrqrqr{$F$}}
\end{pspicture*}
\end{center}

\end{multicols}


To prove that two triangles are congruent, one can use one of the following methods :
\begin{multicols}{2}
\null
\vspace{0.5cm}
\begin{description}
\item[SSS :] Prove that the triangles have three pairs of equal sides.\\
\item[SAS :] Prove that the triangles have two pairs of equal sides and the angles between these sides equal.\\
\item[ASA :] Prove that the triangles have two pairs of equal angles and the sides between these sides equal.
\end{description}

\columnbreak
\psset{unit=0.75cm} \pspicture(0,-0.5)(9,2.5)
\pspolygon(0,0)(4,0)(3,2)
\psline(1.9,0.1)(2.1,-0.1)
\pscircle(1.5,1){0.1}
\psline(3.35,1.05)(3.55,1.15)
\psline(3.4,0.95)(3.6,1.05)
\pspolygon(5,0)(9,0)(8,2)
\psline(6.9,0.1)(7.1,-0.1)
\pscircle(6.5,1){0.1}
\psline(8.35,1.05)(8.55,1.15)
\psline(8.4,0.95)(8.6,1.05)
\endpspicture

\psset{unit=0.75cm} \pspicture(0,-0.5)(9,2.5)
\pspolygon(0,0)(4,0)(3,2)
\psline(1.9,0.1)(2.1,-0.1)
\pscircle(1.5,1){0.1}
\psarc(0,0){0.5}{0}{32}
\pspolygon(5,0)(9,0)(8,2)
\psline(6.9,0.1)(7.1,-0.1)
\pscircle(6.5,1){0.1}
\psarc(5,0){0.5}{0}{32}
\endpspicture

\psset{unit=0.75cm} \pspicture(0,-0.5)(9,2.5)
\pspolygon(0,0)(4,0)(3,2)
\psline(1.9,0.1)(2.1,-0.1)
\psarc(0,0){0.5}{0}{32}
\psarc(4,0){0.5}{120}{180}
\pspolygon(5,0)(9,0)(8,2)
\psline(6.9,0.1)(7.1,-0.1)
\psarc(5,0){0.5}{0}{32}
\psarc(9,0){0.5}{120}{180}
\endpspicture

\end{multicols}

\pagebreak

\moddocdis{Congruent triangles}{01}{AP09}{Exercises}

\sectionblack{Exercise 1}

Let $ABCD$ be a parallelogram.
\begin{enumerate}
\item Prove that triangles $ABC$ and $CDA$ are congruent.
\item Find another pair of congruent triangles on this figure.
\item Let $O$ be the intersection point of the diagonals of $ABCD$.
Find two pairs of congruent triangles involving the point $O$. Prove these properties.
\end{enumerate}

\sectionblack{Exercise 2}

Let $ABC$ be an isosceles triangle with main vertex $B$, and $H$ the foot of the altitude through $B$. Find two
congruent triangles in the figure and prove the property.

\sectionblack{Exercise 3}

Let $ABCD$ be an isosceles trapezium, where
 \begin{itemize}
 \itemb the sides $AD$ and $BC$ are parallel ;
 \itemb the sides $AB$ and $CD$ are equal ;
 \itemb the angles $ABC$ and $BCD$ are equal.
 \end{itemize}
Let $I$ be the midpoint of the side $BC$.
\begin{enumerate}
\item Prove that the triangles $ABI$ and $CDI$ are congruent.
\item Deduce a property about triangle $ADI$.
\end{enumerate}

\sectionblack{Exercise 4}

Let $ABC$ be an equilateral triangle, with $M$, $N$, $P$ three
points respectively on the sides $BC$, $CA$ and $AB$ such that
$BM=CN=AP$.
\begin{enumerate}
\item Prove that the triangles $BMP$, $CNM$ and $APN$ are congruent
to one another.
\item Deduce a property about triangle $MNP$.
\end{enumerate}

\pagebreak

\moddocdis{Congruent triangles}{01}{AP09}{Answers}


\sectionblack{Exercise 1}
\corrige
\begin{minipage}{10cm}
\begin{enumerate}
\item \begin{itemize}
 \item $AB=DC$ because $ABCD$ is a parallelogram.
 \item $BC=DA$ because $ABCD$ is a parallelogram.
 \item $AC=AC$ because it's a commun side.
\end{itemize}
so the triangles $ABC$ and $CDA$ have three pairs of equal sides, so they are congruent.
\end{enumerate}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
\psset{xunit=0.8cm,yunit=0.8cm}
\begin{pspicture*}(0,0)(6,4)
\psset{dotstyle=*,dotsize=3pt 0}
\pspolygon(1,1)(4,1)(5,3)(2,3)
\psdots(1,1)\rput[bl](0.7,1.12){$A$}
\psdots(4,1)\rput[bl](4.24,1.12){$B$}
\psdots(5,3)\rput[bl](5.14,3.12){$C$}
\psdots(2,3)\rput[bl](1.7,3.12){$D$}
\psdots(3,2)\rput[bl](3.08,2.12){$O$}
\pspolygon[linestyle=dashed,linecolor=red](5,3)(4,1)(3,2)
\pspolygon[linestyle=dashed,linecolor=red](1,1)(2,3)(3,2)
\end{pspicture*}
\end{center}
\end{minipage}

\begin{enumerate}\setcounter{enumi}{01}
\item The triangles $ABD$ and $BCD$ are congruent too.
\item \begin{itemize}
 \item $AO=OC$ because $O$ is the midpoint of the diagonals.
 \item $DO=OB$ because $O$ is the midpoint of the diagonals.
 \item $AD=BC$ because $ABCD$ is a parallelogram.
\end{itemize}
so the triangles $AOD$ and $COB$ have three pairs of equal sides, so they are congruent.
\begin{itemize}
 \item $AO=OC$ because $O$ is the midpoint of the diagonals.
 \item $DO=OB$ because $O$ is the midpoint of the diagonals.
 \item $AB=DC$ because $ABCD$ is a parallelogram.
\end{itemize}
so the triangles $AOB$ and $COD$ have three pairs of equal sides, so they are congruent.
\end{enumerate}

\sectionblack{Exercise 2}

\begin{minipage}{10cm}
\begin{itemize}
\item $BH=BH$ because it's a common side.
\item $AB=CB$ because $\triangle ABC$ is isosceles.
\item $\angle BAH=\angle BCH$ because $\triangle ABC$ is isosceles.
\end{itemize}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
\psset{xunit=0.75cm,yunit=0.75cm}
\begin{pspicture*}(-1,-1)(5,3)
\psset{dotstyle=*,dotsize=3pt 0}
\pspolygon(0,0)(4,0)(2,2)
\psdots(0,0)\rput[tr](0,-0.2){$A$}
\psdots(4,0)\rput[tl](4,-0.2){$C$}
\psdots(2,2)\rput[bl](2,2){$B$}
\psdots(2,0)\rput[t](2,-0.2){$H$}
\pspolygon[linestyle=dashed,linecolor=blue](0,0)(2,0)(2,2)
\pspolygon[linestyle=dashed,linecolor=red](4,0)(2,0)(2,2)
\end{pspicture*}
\end{center}
\end{minipage}
So the triangles $ABC$ and $CDA$ have two pairs of equal sides and the angles between these sides
equal, so they are congruent.

\sectionblack{Exercise 3}

\begin{minipage}{10cm}
\begin{enumerate}
\item \begin{itemize}
      \item $CI=BI$ because $I$ is the midpoint of $BC$.
      \item $AB=CD$ because $ABCD$ is an isosceles trapezium.
      \item $\angle ABI=\angle ICD$ because $\triangle ABC$ is isosceles.
      \end{itemize}
So $\triangle ABC$ and $\triangle CDA$ have two pairs of equal sides and the angles between these
sides equal, so they are congruent.
\end{enumerate}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
\psset{xunit=0.75cm,yunit=1.5cm}
\begin{pspicture*}(-1,-1)(6,2)
\psset{dotstyle=*,dotsize=3pt 0}
\pspolygon(0,0)(1,1)(4,1)(5,0)
\psdots(0,0)\uput[dl](0,0){$A$}
\psdots(1,1)\uput[ul](1,1){$B$}
\psdots(4,1)\uput[ur](4,1){$C$}
\psdots(5,0)\uput[dr](5,0){$D$}
\psdots(2.5,1)\uput[u](2.5,1){$I$}
\pspolygon[linestyle=dashed,linecolor=blue](0,0)(1,1)(2.5,1)
\pspolygon[linestyle=dashed,linecolor=red](5,0)(4,1)(2.5,1)
\end{pspicture*}
\end{center}
\end{minipage}

\begin{enumerate}\setcounter{enumi}{01}
\item Since the triangles $ABC$ and $CDA$ are congruent, the third sides are also equal so $AI=ID$.
Consequently $\triangle ADI$ is isosceles.
\end{enumerate}

\sectionblack{Exercise 4}

\begin{minipage}{10cm}
\begin{enumerate}
\item \begin{itemize}
      \item $BM=CN$ according to the instructions.
      \item $BP=CM$ because $BP=side-AP=side-BM=CM$.
      \item $\angle PBM=\angle MCN$ because the triangle $ABC$ is equilateral.
      \end{itemize}
      So $\triangle BMP$ and $\triangle CNM$ have two pairs of equal sides and the angles between these sides
equal, so they are congruent.
\end{enumerate}
\end{minipage}
\begin{minipage}{6cm}
 \begin{center}
      \psset{xunit=0.75cm,yunit=0.75cm}
      \begin{pspicture*}(-1,-1)(6,4)
      \psset{dotstyle=*,dotsize=3pt 0}
      \pspolygon(0,0)(3,0)(1.5,2.6)
      \psdots(0,0)\uput[dl](0,0){$A$}
      \psdots(3,0)\uput[dr](3,0){$B$}
      \psdots(1.5,2.6)\uput[u](1.5,2.6){$C$}
      \psdots(2.7,0.52)\uput[ur](2.7,0.52){$M$}
      \psdots(1.2,2.08)\uput[ul](1.2,2.08){$N$}
      \psdots(0.6,0)\uput[d](0.6,0){$P$}
      \pspolygon[linestyle=dashed,linecolor=red](3,0)(2.7,0.52)(0.6,0)
      \pspolygon[linestyle=dashed,linecolor=blue](1.5,2.6)(2.7,0.52)(1.2,2.08)
      \end{pspicture*}
      \end{center}\end{minipage}
\begin{enumerate}\setcounter{enumi}{01}
\item In the same way, we prove that $\triangle CNM$ and $\triangle APN$ are congruent.
\item We can deduce that the third sides of the triangles are equal, so $\triangle MNP$ is equilateral.
\end{enumerate}


\end{document}
