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Revision difference : corresponding angles in transversal cutting |
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Version 2 |
| \PMlinkescapeword{cut} |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-3,-3)(3,3) |
\begin{pspicture}(-3,-3)(3,3) |
| \rput[b](-3,-3){.} |
\rput[b](-3,-3){.} |
| \rput[a](3,3){.} |
\rput[a](3,3){.} |
| \psline(-3,-3)(3,3) |
\psline(-3,-3)(3,3) |
| \psline(-1,-3)(5,2) |
\psline(-1,-3)(5,2) |
| \psline(-2,2)(5,-2) |
\psline(-2,2)(5,-2) |
| \rput[r](1.12,0.65){$\alpha$} |
\rput[r](1.12,0.65){$\alpha$} |
| \rput[r](2.68,-0.3){$\beta$} |
\rput[r](2.68,-0.3){$\beta$} |
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\rput[l](3.1,3){$\ell$}
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\rput[l](3.1,3){$\l$}
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| \rput[l](5.2,2){$m$} |
\rput[l](5.2,2){$m$} |
| \rput[r](-2.2,2){$t$} |
\rput[r](-2.2,2){$t$} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| The following theorem is valid in Euclidean geometry: |
\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. |
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| \textbf{Theorem.}\, If two lines ($\ell$ and $m$) are cut by a third line, called a {\em transversal} ($t$), and one pair of corresponding angles (\PMlinkname{e.g.}{Eg} $\alpha$ and $\beta$) are congruent, then the cut lines are parallel. |
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| Its converse theorem is also valid in Euclidean geometry: |
There is also valid the converse |
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\textbf{Theorem.}\, If two parallel lines ($\ell$ and $m$) are cut by a transversal ($t$), then each pair of corresponding angles (e.g. $\alpha$ and $\beta$) are congruent.
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\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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