document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179597>Question 1179597</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Formulate a simple geometric statement that can be proven using <BR>\n" );
document.write( "any combination of the following: congruence axioms, substitution, and <BR>\n" );
document.write( "definitions. Show two different ways to prove this statement. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850209 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=850209\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Okay, let's formulate a simple geometric statement and prove it in two different ways using congruence axioms, substitution, and definitions.\r<BR>\n" );
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document.write( "**Geometric Statement:**\r<BR>\n" );
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document.write( "**Given:** Two line segments AB and CD, where AB ≅ CD. Let E be a point on AB such that AE ≅ EB. Let F be a point on CD such that CF ≅ FD.\r<BR>\n" );
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document.write( "**Prove:** AE ≅ CF.\r<BR>\n" );
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document.write( "**Proof 1: Using Substitution and Congruence**\r<BR>\n" );
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document.write( "| Statement | Reason |<BR>\n" );
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document.write( "| 1. AB ≅ CD | Given |<BR>\n" );
document.write( "| 2. AE ≅ EB | Given |<BR>\n" );
document.write( "| 3. CF ≅ FD | Given |<BR>\n" );
document.write( "| 4. AE = EB | Definition of congruent segments (If segments are congruent, their lengths are equal) |<BR>\n" );
document.write( "| 5. CF = FD | Definition of congruent segments |<BR>\n" );
document.write( "| 6. AB = CD | Definition of congruent segments |<BR>\n" );
document.write( "| 7. AE + EB = AB | Segment Addition Postulate |<BR>\n" );
document.write( "| 8. CF + FD = CD | Segment Addition Postulate |<BR>\n" );
document.write( "| 9. AE + AE = AB | Substitution (Substitute AE for EB in step 7) |<BR>\n" );
document.write( "| 10. CF + CF = CD | Substitution (Substitute CF for FD in step 8) |<BR>\n" );
document.write( "| 11. 2AE = AB | Simplification |<BR>\n" );
document.write( "| 12. 2CF = CD | Simplification |<BR>\n" );
document.write( "| 13. 2AE = CD | Substitution (Substitute CD for AB in step 11, using step 6) |<BR>\n" );
document.write( "| 14. 2AE = 2CF | Substitution (Substitute 2CF for CD in step 13, using step 12) |<BR>\n" );
document.write( "| 15. AE = CF | Division Property of Equality (Divide both sides by 2) |<BR>\n" );
document.write( "| 16. AE ≅ CF | Definition of congruent segments |\r<BR>\n" );
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document.write( "**Proof 2: Using a Different Substitution Approach**\r<BR>\n" );
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document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. AB ≅ CD | Given |<BR>\n" );
document.write( "| 2. AE ≅ EB | Given |<BR>\n" );
document.write( "| 3. CF ≅ FD | Given |<BR>\n" );
document.write( "| 4. AE = EB | Definition of congruent segments |<BR>\n" );
document.write( "| 5. CF = FD | Definition of congruent segments |<BR>\n" );
document.write( "| 6. AB = CD | Definition of congruent segments |<BR>\n" );
document.write( "| 7. AE + EB = AB | Segment Addition Postulate |<BR>\n" );
document.write( "| 8. CF + FD = CD | Segment Addition Postulate |<BR>\n" );
document.write( "| 9. AB/2 = AE | Division Property of Equality and Substitution(Using step 4, 7, and the fact that since AE=EB, AB is 2AE) |<BR>\n" );
document.write( "| 10. CD/2 = CF | Division Property of Equality and Substitution(Using step 5, 8, and the fact that since CF=FD, CD is 2CF) |<BR>\n" );
document.write( "| 11. AB/2 = CD/2 | Division Property of Equality (Dividing both sides of step 6 by 2) |<BR>\n" );
document.write( "| 12. AE = CD/2 | Substitution (Substitute AE for AB/2 in step 11, using step 9) |<BR>\n" );
document.write( "| 13. AE = CF | Substitution(Substitute CF for CD/2 in step 12, using step 10)|<BR>\n" );
document.write( "| 14. AE ≅ CF | Definition of congruent segments |\r<BR>\n" );
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document.write( "**Explanation of the Proofs:**\r<BR>\n" );
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document.write( "* Both proofs rely on the fundamental definitions of congruent segments, which state that if segments are congruent, their lengths are equal, and vice versa.<BR>\n" );
document.write( "* The Segment Addition Postulate is used to express the lengths of the given segments in terms of their parts.<BR>\n" );
document.write( "* The proofs then use substitution to manipulate the equations and arrive at the desired conclusion.<BR>\n" );
document.write( "* The division property of equality is used to isolate the segment lengths.\r<BR>\n" );
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document.write( "These proofs are simple yet demonstrate the power of combining definitions, postulates, and substitution to prove geometric statements.<BR>\n" );
document.write( " </SPAN>\n" );
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