document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210469>Question 1210469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the following conditions ensures that triangles ABC and DEF are similar?  Select all that apply.\r<BR>\n" );
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document.write( "(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE\r<BR>\n" );
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document.write( "(2) A = D, AB/EF = BC/DE\r<BR>\n" );
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document.write( "(3) AB perp CD, DE perp AE, AB = DE and BC = BD\r<BR>\n" );
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document.write( "(4) AB parallel to BC, AB parallel to AC, CA parallel to FD </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852804 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53765)</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=852804\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Which  of the following conditions ensures that triangles ABC and DEF are similar? Select all that apply.\r<BR>\n" );
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document.write( "(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE<BR>\n" );
document.write( "(2) A = D, AB/EF = BC/DE<BR>\n" );
document.write( "(3) AB perp CD, DE perp AE, AB = DE and BC = BD<BR>\n" );
document.write( "(4) AB parallel to BC, AB parallel to AC, CA parallel to FD<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The analysis by &nbsp;@CPhill in part &nbsp;(1) &nbsp;is incorrect and his conclusion \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for this part is incorrect, &nbsp;too.\r<BR>\n" );
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document.write( "Indeed,  A = B means that triangle ABC is isosceles with congruent angles A and B at its base.\r\n" );
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document.write( "Next,    F = E means that triangle DEF is isosceles with congruent angles F and E at its base.\r\n" );
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document.write( "The condition D = C means that the angles D and C in these two triangles, ABC and DEF,\r\n" );
document.write( "opposite to their bases AB and FE, are congruent.\r\n" );
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document.write( "So, the triangles ABC and DEF are similar due to the AAA-test.\r\n" );
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document.write( "That is true that the vertices of these triangles are listed in non-canonic order.\r\n" );
document.write( "The canonic order should be consistent, like ABC and EFD,\r\n" );
document.write( "but this does not interfere for triangles ABC and DEF  (or ABC and EFD) to be similar.\r\n" );
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document.write( "The last imposed condition in (1), AB/AD = BC/BE is not relevant to their similarity,\r\n" );
document.write( "but does not contradict to it.\r\n" );
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document.write( "The similarity of triangles ABC and FED is provided by the conditions on congruency their corresponding angles.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "So again, &nbsp;in part &nbsp;(1), &nbsp;the analysis and the conclusion by @CPhill are incorrect.\r<BR>\n" );
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