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\lhead{\color{blue} \large Lecture Notes}
\chead{\color{black} \Large  Area}
\rhead{ page   \ \thepage}
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\lfoot{\small   \copyright $\;$  Hidegkuti,  Powell,  2016}
\rfoot{\small Last revised:  October    3, 2016}
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\begin{center}
{\LARGE Part 4 - Parallelograms\bigskip }
\end{center}

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\textbf{Definition: }\ A parallelogram is a four sided polygon with two
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\bigskip

It is a proven fact that the opposite sides of a parallelogram are of equal
length. \ This is not part of the definition, but it is an important
property that we need to remember. \ Also, we could prove that the diagonals
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Another important property of parallelograms is the connection between its
angles. \ In every parallelogram, the opposite angles are equal, and the two
angles along each side add up to $180^{\circ }$. \ We call two such angles
supplemental.

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us to easily compute the area of the parallelogram.\bigskip

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\textbf{Theorem:\ }The area of a parallelogram with sides $a,$ $b$ and
height $h$ belonging to $a$ is $A=ah$.\FRAME{dtbpF}{1.6034in}{0.966in}{0pt}{%
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\pagebreak

Proof: We will use (surprise, surprise!) a previously proven result. If we
cut off a triangle and paste it back as show on the picture below, we obtain
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rectangle with sides $a$ and $h$.\bigskip

\textit{Example 4}: \ Find the area of the parallelogram shown on the
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"XNPEU";}}Solution: \ We apply the formula for the area of a parallelogram.%
\begin{equation*}
A=ah=5\unit{cm}\left( 3\unit{cm}\right) =15\unit{cm}^{2}
\end{equation*}

\bigskip

\bigskip \bigskip \bigskip

\begin{center}
{\LARGE Practice Problems\bigskip }
\end{center}

\begin{enumerate}
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\pagebreak

\item Find the values of $x$ and $y$ if it is given that the parallelogram
shown on the picture has perimeter of $26\unit{ft}$ and area $24\unit{ft}%
^{2} $.\FRAME{dtbpF}{2.1248in}{1.3681in}{0pt}{}{}{pic9.bmp}{\special%
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"XNPEU";}}\bigskip \bigskip \bigskip
\end{enumerate}

\begin{center}
{\Large Practice Problems - Answers}\bigskip
\end{center}

\begin{enumerate}
\item $A=21\unit{ft}^{2}$

\item $A=60\unit{in}^{2}$

\item $x=5\unit{ft}$ \ \ \ \ $y=3\unit{ft}$\vspace{0.8in}
\end{enumerate}

\vspace{4in}

\href{http://www.teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html%
}{For more documents like this, visit our page at\
http://www.teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail
questions or comments to mhidegkuti@ccc.edu.}

\end{document}