document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177180>Question 1177180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given: Segment NK is a median of triangle JMN; IN>NM Prove: m<1>m<2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850508 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850508\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Given Information**\r<BR>\n" );
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document.write( "* NK is a median of triangle JMN. This means that NK bisects side JM, so JK = KM.<BR>\n" );
document.write( "* IN > NM\r<BR>\n" );
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document.write( "**2. Triangle Inequality Theorem**\r<BR>\n" );
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document.write( "In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.\r<BR>\n" );
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document.write( "**3. Applying the Triangle Inequality Theorem to Triangle INK**\r<BR>\n" );
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document.write( "* IN + NK > IK \r<BR>\n" );
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document.write( "**4. Applying the Triangle Inequality Theorem to Triangle NMK**\r<BR>\n" );
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document.write( "* NM + NK > MK\r<BR>\n" );
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document.write( "**5. Using the Given Information and Steps 3 and 4**\r<BR>\n" );
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document.write( "* Since IN > NM (given), and NK is common to both triangles INK and NMK, we can conclude that:<BR>\n" );
document.write( "   IN + NK > NM + NK<BR>\n" );
document.write( "* This further implies that IK > MK\r<BR>\n" );
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document.write( "**6. Relating IK and MK to JK and KM**\r<BR>\n" );
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document.write( "* We know that JK = KM (because NK is a median).<BR>\n" );
document.write( "* Therefore, IK > JK\r<BR>\n" );
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document.write( "**7. Hinge Theorem**\r<BR>\n" );
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document.write( "The Hinge Theorem states that if two triangles have two congruent sides, then the triangle with the larger included angle has the longer third side.\r<BR>\n" );
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document.write( "**8. Applying the Hinge Theorem**\r<BR>\n" );
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document.write( "* In triangles JNK and KNM:<BR>\n" );
document.write( "    * JN = NM (given)<BR>\n" );
document.write( "    * NK = NK (common side)<BR>\n" );
document.write( "    * IK > JK (from step 6)<BR>\n" );
document.write( "* Therefore, by the Hinge Theorem, m∠1 > m∠2.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "We have successfully proven that m∠1 > m∠2 using the given information, the Triangle Inequality Theorem, and the Hinge Theorem. </SPAN>\n" );
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