document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195016>Question 1195016</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a parallelogram □ABCD, with AD > AB.<BR>\n" );
document.write( "The bisector of ∠A intersects 𝐵𝐶 at G, and the bisector of ∠B intersects𝐴𝐷 at H.<BR>\n" );
document.write( "Prove that □ABGH is a rhombus. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827369 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=827369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This is one way to draw out the diagram to supplement the answer @greenestamps posted<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/ygmS14C.png\"><BR>\n" );
document.write( "The 90 degree angle can be found using the scratch work shown at the bottom of the solution page.<BR>\n" );
document.write( "We must have AD be longer than AB, otherwise the points G and H wouldn't be possible.\r<BR>\n" );
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document.write( "It then leads to this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/iVdhQHi.png\"><BR>\n" );
document.write( "Note that if x+y = 90 and y+z = 90, then we can say x = z.<BR>\n" );
document.write( "This line of thinking helps determine the missing blue and red angles.\r<BR>\n" );
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document.write( "This shows we have four congruent right triangles.<BR>\n" );
document.write( "Each hypotenuse is the same, hence all four sides of this sub-figure are the same length.<BR>\n" );
document.write( "This proves ABGH is a rhombus.\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Scratch Work:<BR>\n" );
document.write( "<font color=blue>Angle HAB</font> + <font color=red>Angle ABG</font> = 180<BR>\n" );
document.write( "<font color=blue>(Angle HAP + Angle PAB)</font> + <font color=red>(Angle ABP + Angle PBG)</font> = 180<BR>\n" );
document.write( "<font color=blue>(Angle PAB + Angle PAB)</font> + <font color=red>(Angle ABP + Angle ABP)</font> = 180<BR>\n" );
document.write( "<font color=blue>2*(Angle PAB)</font> + <font color=red>2*(Angle ABP)</font> = 180<BR>\n" );
document.write( "2*(<font color=blue>Angle PAB</font> + <font color=red>Angle ABP</font>) = 180<BR>\n" );
document.write( "<font color=blue>Angle PAB</font> + <font color=red>Angle ABP</font> = 180/2<BR>\n" );
document.write( "<font color=blue>Angle PAB</font> + <font color=red>Angle ABP</font> = 90<BR>\n" );
document.write( "Hence, triangle APB is a right triangle. This means all angles around point P are 90 degree angles as well.<BR>\n" );
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