document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=846102>Question 846102</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.<BR>\n" );
document.write( "How do I solve this proof? I know that I can start with trapezoid ABCD (given), but I'm unsure of how to go on. I appreciate any help  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #509612 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=509612\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose that in trapezoid <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large ABCD\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large AB || CD\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large AC = BD\">. Denote by E the intersection of AC and BD.\r<BR>\n" );
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document.write( "Because triangles ABE and CDE are similar by angle-angle-angle, we can write DE = x, CE = y, then AE = ky and BE = kx. Since AC = BD, we have ky + y = kx + x --> y(k+1) = x(k+1), which implies x = y. Therefore DE = CE and AE = BE.\r<BR>\n" );
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document.write( "It follows that triangles ABE and CDE are isosceles, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \angle CAB = \angle DBA\">, and by SAS, triangles ABC and BAD are congruent, so AD = BC. The trapezoid is isosceles. </SPAN>\n" );
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