document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206126>Question 1206126</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove by contradiction that the diagonals of a kite intersect at right angles? My proof is very wordy,can you help me be more precise. Thank you </SPAN>\n" );
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document.write( "Consider kite ABCD.<BR>\n" );
document.write( "By definition of a kite, adjacent sides are congruent so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB+=+BC ALIGN=MIDDLE  ALT=\"AB+=+BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD+=+DA ALIGN=MIDDLE  ALT=\"CD+=+DA\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB+%3C%3E+CD ALIGN=MIDDLE  ALT=\"AB+%3C%3E+CD\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Not all 4 sides are congruent, otherwise we would have a rhombus.\r<BR>\n" );
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document.write( "One key property of a kite is that one diagonal is cut in half by the other. Both diagonals are not bisected (otherwise we would have a parallelogram). Let's consider diagonal AC is bisected to get AE = EC where E is the intersection of the diagonals.\r<BR>\n" );
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document.write( "Proof by Contradiction:<BR>\n" );
document.write( "Assume the diagonals do not intersect at right angles.<BR>\n" );
document.write( "It means neither angle AEB nor BEC is 90 degrees and furthermore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C2%2C%22angle%22%2C%22AEB%22%29+%3C%3E+matrix%281%2C2%2C%22angle%22%2C%22BEC%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So either,<BR>\n" );
document.write( "angle AEB > angle BEC<BR>\n" );
document.write( "or<BR>\n" );
document.write( "angle AEB < angle BEC\r<BR>\n" );
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document.write( "If angle AEB > angle BEC, then AB > BC due to the Hinge Theorem (see links below).<BR>\n" );
document.write( "But this contradicts AB = BC.\r<BR>\n" );
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document.write( "If angle AEB < angle BEC, then AB < BC due to the Hinge Theorem which also contradicts AB = BC\r<BR>\n" );
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document.write( "Either contradiction leads us to conclude that the opposite of the assumption above must be the case. Therefore, we conclude that the diagonals of a kite are perpendicular.\r<BR>\n" );
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document.write( "Further reading about the Hinge Theorem<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"http://ceemrr.com/Geometry1/HingeTheorem/HingeTheorem_print.html\">http://ceemrr.com/Geometry1/HingeTheorem/HingeTheorem_print.html</a><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://mathbitsnotebook.com/Geometry/SegmentsAnglesTriangles/SATHinge.html\">https://mathbitsnotebook.com/Geometry/SegmentsAnglesTriangles/SATHinge.html</a><BR>\n" );
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