document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1075229>Question 1075229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with the following proof. Thank you!\r<BR>\n" );
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document.write( "How to prove statement \"If a rectangle is a square, then its diagonals are perpendicular\" and its converse \" If the diagonals of a rectangle are perpendicular, then the rectangle is a square\"  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689972 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=689972\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If a rectangle is a square,<BR>\n" );
document.write( "then all its 4 sides are congruent.<BR>\n" );
document.write( "Each of its diagonals splits the square into<BR>\n" );
document.write( "two isosceles right triangles.<BR>\n" );
document.write( "Each one is a right triangle,<BR>\n" );
document.write( "because it has a right angle from the rectangle/square,<BR>\n" );
document.write( "and it is isosceles because it has two congruent legs<BR>\n" );
document.write( "(that are two congruent adjacent sides of the square).<BR>\n" );
document.write( "Then, each of those isosceles right triangles has two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > base angles.<BR>\n" );
document.write( "Those are the angles between the diagonals and the sides of the square.<BR>\n" );
document.write( "together, the two diagonals split the square into four triangles.<BR>\n" );
document.write( "Each of those four triangles has two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles,<BR>\n" );
document.write( "adjacent to the side that is a side of the square.<BR>\n" );
document.write( "The remaining angle, the one between the two sides along diagonals of the square,<BR>\n" );
document.write( "must be a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo-45%5Eo-45%5Eo=90%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo-45%5Eo-45%5Eo=90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle,<BR>\n" );
document.write( "so the diagonals are perpendicular.<BR>\n" );
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document.write( "If the diagonals of a rectangle are perpendicular,<BR>\n" );
document.write( "then they split the rectangle into 4 right triangles.<BR>\n" );
document.write( "The diagonals of a parallelogram (including a rectangle) bisect each other.<BR>\n" );
document.write( "If we call the diagonals' lengths <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b ALIGN=MIDDLE  ALT=\"2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we can say that each of those 4 right triangles has,<BR>\n" );
document.write( "one leg of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and one leg of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those 4 right triangles must be congruent,<BR>\n" );
document.write( "meaning that their hypotenuses must be congruent,<BR>\n" );
document.write( "and since those hypotenuses are sides of the rectangle,<BR>\n" );
document.write( "the rectangle has 4 congruent sides,<BR>\n" );
document.write( "which makes it a square.\r<BR>\n" );
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