document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102508>Question 1102508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> n the figure triangle abc is congruent to triangle bch which is congruent to triangle hgb is it possible to determine angle gbh why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717252 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717252\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When we say that triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28ABC%29 ALIGN=MIDDLE  ALT=\"red%28ABC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is congruent to triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28BCH%29 ALIGN=MIDDLE  ALT=\"green%28BCH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we are saying about the relations of side length that.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AB%29=green%28BC%29 ALIGN=MIDDLE  ALT=\"red%28AB%29=green%28BC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28BC%29=green%28CH%29 ALIGN=MIDDLE  ALT=\"red%28BC%29=green%28CH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AC%29=green%28BH%29 ALIGN=MIDDLE  ALT=\"red%28AC%29=green%28BH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case, because side BC is shared between two triangles,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AB%29=green%28BC%29=red%28BC%29=green%28CH%29 ALIGN=MIDDLE  ALT=\"red%28AB%29=green%28BC%29=red%28BC%29=green%28CH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AB%29=red%28BC%29 ALIGN=MIDDLE  ALT=\"red%28AB%29=red%28BC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28BC%29=green%28CH%29 ALIGN=MIDDLE  ALT=\"green%28BC%29=green%28CH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "That tells us that both triangles are at least isosceles,<BR>\n" );
document.write( "and may even be a peculiar type of isosceles triangle<BR>\n" );
document.write( "that we call an equilateral triangle.<BR>\n" );
document.write( "I can also picture those two triangles looking like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C350%2C-0.5%2C5.5%2C-3.5%2C3.5%2C%0D%0Atriangle%280%2C0%2C5%2C0%2C1%2C3%29%2Ctriangle%280%2C0%2C5%2C0%2C4%2C-3%29%2C%0D%0Alocate%28-0.2%2C0.15%2CB%29%2Clocate%285.1%2C0.15%2CC%29%2C%0D%0Alocate%280.9%2C3.3%2CH%29%2Clocate%283.9%2C-3%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , or like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-0.5%2C5.5%2C-0.5%2C3.5%2C%0D%0Atriangle%280%2C0%2C5%2C0%2C1%2C3%29%2Ctriangle%280%2C0%2C5%2C0%2C4%2C3%29%2C%0D%0Alocate%28-0.2%2C0.15%2CB%29%2Clocate%285.1%2C0.15%2CC%29%2C%0D%0Alocate%280.9%2C3.3%2CH%29%2Clocate%283.9%2C3.3%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Without comparing sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28BCH%29 ALIGN=MIDDLE  ALT=\"green%28BCH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28HGB%29 ALIGN=MIDDLE  ALT=\"blue%28HGB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we already know that they are both isosceles triangles,<BR>\n" );
document.write( "with vertex marked by the middle letter and flanked by congruent sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28BC%29=green%28CH%29=blue%28HG%29=blue%28GB%29 ALIGN=MIDDLE  ALT=\"green%28BC%29=green%28CH%29=blue%28HG%29=blue%28GB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Angle HGB is the vertex angle of the isosceles triangle,<BR>\n" );
document.write( "and angle GBH is one of the two congruent base angles.<BR>\n" );
document.write( " <BR>\n" );
document.write( "We have three congruent isosceles triangles, looking like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28450%2C350%2C-4.5%2C5.5%2C-3.5%2C3.5%2C%0D%0Atriangle%280%2C0%2C5%2C0%2C1%2C3%29%2Ctriangle%280%2C0%2C5%2C0%2C4%2C-3%29%2C%0D%0Atriangle%280%2C0%2C1%2C3%2C-4%2C3%29%2Clocate%28-4.1%2C3.3%2CG%29%2C%0D%0Alocate%28-0.15%2C0%2CB%29%2Clocate%285.1%2C0.15%2CC%29%2C%0D%0Alocate%280.9%2C3.3%2CH%29%2Clocate%283.9%2C-3%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28450%2C200%2C-4.5%2C5.5%2C-0.5%2C3.5%2C%0D%0Atriangle%280%2C0%2C5%2C0%2C1%2C3%29%2Ctriangle%280%2C0%2C5%2C0%2C4%2C3%29%2C%0D%0Atriangle%280%2C0%2C1%2C3%2C-4%2C3%29%2Clocate%28-4.1%2C3.3%2CG%29%2C%0D%0Alocate%28-0.15%2C0%2CB%29%2Clocate%285.1%2C0.15%2CC%29%2C%0D%0Alocate%280.9%2C3.3%2CH%29%2Clocate%283.9%2C3.3%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28550%2C180%2C-27.5%2C27.5%2C-9%2C9%2C%0D%0Atriangle%280%2C0%2C25%2C0%2C1%2C7%29%2Ctriangle%280%2C0%2C25%2C0%2C24%2C-7%29%2C%0D%0Atriangle%280%2C0%2C1%2C7%2C-24%2C7%29%2Clocate%28-25.5%2C7.5%2CG%29%2C%0D%0Alocate%28-0.75%2C0%2CB%29%2Clocate%2825.5%2C1%2CC%29%2C%0D%0Alocate%280.5%2C8.5%2CH%29%2Clocate%2824%2C-7%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28550%2C180%2C-27.5%2C27.5%2C-9%2C9%2C%0D%0Atriangle%280%2C0%2C25%2C0%2C1%2C7%29%2Ctriangle%280%2C0%2C25%2C0%2C24%2C7%29%2C%0D%0Atriangle%280%2C0%2C1%2C7%2C-24%2C7%29%2Clocate%28-25.5%2C7.5%2CG%29%2C%0D%0Alocate%28-0.75%2C0%2CB%29%2Clocate%2825.5%2C1%2CC%29%2C%0D%0Alocate%280.5%2C8.5%2CH%29%2Clocate%2824%2C8.5%2CA%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we do not know the measure of their angles.<BR>\n" );
document.write( " <BR>\n" );
document.write( "It could be that they are equilateral,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AC%29=green%28BH%29=all ALIGN=MIDDLE  ALT=\"red%28AC%29=green%28BH%29=all\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=the ALIGN=MIDDLE  ALT=\"the\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=other ALIGN=MIDDLE  ALT=\"other\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=side ALIGN=MIDDLE  ALT=\"side\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lengths ALIGN=MIDDLE  ALT=\"lengths\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and all the angles measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, without any more information<BR>\n" );
document.write( "(which could be implied by the figure I do not see),<BR>\n" );
document.write( "I do not know if they are equilateral triangles,<BR>\n" );
document.write( "and I do not know the angles' measures. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );