document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175958>Question 1175958</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram, AB || DG and AD || FG. If AB = 3, BC = 2, AC = 4, CE = 4, and EF = 5, what is the sum of the measures of the sides of the triangles? Express your answer as a common fraction. \r<BR>\n" );
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document.write( "My thoughts, the once that are parallel are equal to each other, so I think the total will be 1/6. \r<BR>\n" );
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document.write( "Please correct me\r<BR>\n" );
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document.write( "Link to the diagram - https://drive.google.com/file/d/1-Vuzfqi9S4GsXXKwbWqOJXlsCFfAuTMl/view?usp=sharing </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #801769 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=801769\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's the deal.\r<BR>\n" );
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document.write( "AD is parallel to FG.<BR>\n" );
document.write( "this makes angle GFE equal to angle DCE because they are alternate interior angles to those two parallel lines.<BR>\n" );
document.write( "those angles are labeled as 2 in my diagram.<BR>\n" );
document.write( "also angle FGE is equal to angle EDC because they are alternate interior angles to the same two parallel lines.<BR>\n" );
document.write( "those angles are labeled as 1 in my diagram.\r<BR>\n" );
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document.write( "AB is parallel to DG.<BR>\n" );
document.write( "this makes angles BAC equal to angle CDE because they are alternate interior angles to those two parallel lines.<BR>\n" );
document.write( "those angles are labeled 1 in my diagram.<BR>\n" );
document.write( "also angle CBA is equal to angle DEC because they are alternate interior angles to those two parallel lines.<BR>\n" );
document.write( "those angles are labeled 3 in my diagram.\r<BR>\n" );
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document.write( "you also have angle BCA = angle DCE because they are vertical angles to each other.<BR>\n" );
document.write( "those two angles are labeled 2 in my diagram.\r<BR>\n" );
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document.write( "you also have angle DEC equal to angle GEF because they are vertical angles to each other.<BR>\n" );
document.write( "those two angles are labeled 3 in my diagram.\r<BR>\n" );
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document.write( "the bottom line is that triangles ABC and DEC and GEF are all similar to each other because their angles are equal.<BR>\n" );
document.write( "since they are similar to each other, their sides are proportional.\r<BR>\n" );
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document.write( "the ratio from triangle DEC to ABC is 2/1.<BR>\n" );
document.write( "the ratio from triangle GEF to ABC is 5/2.<BR>\n" );
document.write( "the ratio from triangle GEF to DEC is 5/4.\r<BR>\n" );
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document.write( "this means that corresponding sides of these triangle will have the same ratio to each other.\r<BR>\n" );
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document.write( "to find the corresponding sides, you look for the sides opposite the angles that are equal to each other.\r<BR>\n" );
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document.write( "example:\r<BR>\n" );
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document.write( "BC is opposite angle 1 in its triangle and CE is opposite angle 1 in its triangle.<BR>\n" );
document.write( "the ratio between those two triangle is 2/1, so if BC is 2 then CE is 4.<BR>\n" );
document.write( "this was one of the givens and was used to get the ratio for the other sides.<BR>\n" );
document.write( "since AB was 3 and is opposite angle 2 in its triangle, then DE is 6 because it is opposite angle 2 in its triangle.\r<BR>\n" );
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document.write( "likewise angle 1 from triangle ABC is equal to angle 1 from triangle GEF.<BR>\n" );
document.write( "the side opposite angle 1 in triangle ABE is equal to 2 and the side opposite angle 1 in triangle GEF is equal to 5.<BR>\n" );
document.write( "this sets the ratio of the sides as 5/2 from triangle GEF to ABC.\r<BR>\n" );
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document.write( "likewise the side opposite angle 1 in triangle DEC is equal to 4 and the side opposite angle 1 in triangle GEF is equal to 5.<BR>\n" );
document.write( "this sets the ratio of sides as 5/4 from triangle GEF to DEC.\r<BR>\n" );
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document.write( "you use those ratios to get the rest of the corresponding sides of each triangle.\r<BR>\n" );
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document.write( "now that you know what all the sides measure, you can get their sum.\r<BR>\n" );
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document.write( "you can add each one up individually to get a sum of 49.5\r<BR>\n" );
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document.write( "or, .....\r<BR>\n" );
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document.write( "you can sum up the sides of triangle ABC to get 9.<BR>\n" );
document.write( "since the ratio of triangle DEC to ABC is 2 to 1, then the sum for triangle DEC is 2/1 * 9 = 18.<BR>\n" );
document.write( "since the ratio of triangle GEF to ABC is 5/2,then the sum for triangle GEF is 5/2 * 9 = 22.5<BR>\n" );
document.write( "9 + 18 + 22.5 = 49.5.<BR>\n" );
document.write( "same answer.\r<BR>\n" );
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document.write( "here's my diagram.\r<BR>\n" );
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document.write( "note that the transversals to parallel lines AB and DG are AD and BE.\r<BR>\n" );
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document.write( "note also that the transversals to parallel lines AD and GF are DG and CF.\r<BR>\n" );
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document.write( "here's my diagram and yours below it.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/022605.jpg\" > </SPAN>\n" );
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