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'corresponding angles in transversal cutting'
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| Title of object: |
corresponding angles in transversal cutting |
| Canonical Name: |
CorrespondingAnglesInTransversalCutting |
| Type: |
Theorem |
| Created on: |
2007-06-13 16:54:45 |
| Modified on: |
2007-06-13 16:54:45 |
| Classification: |
msc:51-01 |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\begin{center}
\begin{pspicture}(-3,-3)(3,3)
\rput[b](-3,-3){.}
\rput[a](3,3){.}
\psline(-3,-3)(3,3)
\psline(-1,-3)(5,2)
\psline(-2,2)(5,-2)
\rput[r](1.12,0.65){$\alpha$}
\rput[r](2.68,-0.3){$\beta$}
\rput[l](3.1,3){$\l$}
\rput[l](5.2,2){$m$}
\rput[r](-2.2,2){$t$}
\end{pspicture}
\end{center}
\textbf{Theorem.}\, If two lines ($l$ and $m$) are cut by a third line, the so-called {\em transversal} ($t$), and one pair of corresponding angles (e.g. $\alpha$ and $\beta$) are congruent, then the cut lines are parallel.
There is also valid the converse
\textbf{Theorem.}\, If two parallel lines ($l$ and $m$) are cut by a transversal line ($t$), then each pair of corresponding angles (e.g. $\alpha$ and $\beta$) are congruent.
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