document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001536>Question 1001536</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given: Line P is parallel to line Q<BR>\n" );
document.write( "Line Q is parallel to line R\r<BR>\n" );
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document.write( "Prove: Line P is parallel to line R </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618700 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52863)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618700\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<TABLE >\n" );
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document.write( "When you have two straight lines in the plane, there are two alternatives:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\r<BR>\n" );
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document.write( "1. These two straight lines intersect (as shown in &nbsp;<B>Figure 1a</B>), &nbsp;and<BR>\n" );
document.write( "2. These two straight lines do not intersect even at extended length<BR>\n" );
document.write( "   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(as shown in &nbsp;<B>Figure 1b</B>).<BR>\n" );
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document.write( "<B>Figure 1a</B>. Intersecting  <BR>\n" );
document.write( "straight lines&nbsp;&nbsp;&nbsp;&nbsp;<BR>\n" );
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document.write( "<B>Figure 1b</B>. Parallel <BR>\n" );
document.write( "straight lines<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Definition</B><BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Two straight lines in a plane are called &nbsp;<B>parallel</B>&nbsp; if they do not intersect \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;even at extended length.\r<BR>\n" );
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document.write( "The next statement is a &nbsp;<B>postulate</B>, &nbsp;that is a statement, which is accepted without proof:<BR>\n" );
document.write( "<BLOCKQUOTE><B>For a given straight line and a point in the plane outside of this straight line there is &nbsp;&nbsp;<BR>\n" );
document.write( "one and only one straight line parallel to the given straight line and passing through the <BR>\n" );
document.write( "given point</B>.</BLOCKQUOTE><BR>\n" );
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document.write( "<B>Figure 2</B>. The unique parallel <BR>\n" );
document.write( "straight line passing through <BR>\n" );
document.write( "the given point out the line<BR>\n" );
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document.write( "<B>Theorem</B><BR>\n" );
document.write( "If a straight line &nbsp;<B>a</B>&nbsp; is parallel to a straight line &nbsp;<B>b</B>&nbsp; and a straight line &nbsp;<B>b</B>&nbsp; is parallel to a straight line &nbsp;<B>c</B>, \r<BR>\n" );
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document.write( "then the straight lines &nbsp;<B>a</B>&nbsp; and &nbsp;<B>c</B>&nbsp; are parallel.\r<BR>\n" );
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document.write( "<B>Proof</B><BR>\n" );
document.write( "Let us suppose that straight lines &nbsp;<B>a</B>&nbsp; and &nbsp;<B>c</B>&nbsp; are not parallel. &nbsp;Then they have an intersection point. &nbsp;Let us denote as &nbsp;<B>C</B>&nbsp; \r<BR>\n" );
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document.write( "this intersection point of the straight lines &nbsp;<B>a</B>&nbsp; and &nbsp;<B>c</B>.\r<BR>\n" );
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document.write( "Then we have two straight lines, &nbsp;<B>a</B>&nbsp; and &nbsp;<B>c</B>, &nbsp;passing through the point &nbsp;<B>C</B>&nbsp; and parallel to the straight line &nbsp;<B>b</B>, &nbsp;which is \r<BR>\n" );
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document.write( "impossible due to the &nbsp;<B>postulate</B>&nbsp; above.\r<BR>\n" );
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document.write( "See the lesson &nbsp;<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>&nbsp; in this site.\r<BR>\n" );
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