document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169211>Question 1169211</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given that ∠FAB≅∠GED and C is the midpoint of AE¯¯¯¯¯, which of the following proves that △ABC≅△EDC?\r<BR>\n" );
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document.write( "A.<BR>\n" );
document.write( "1.  ∠FAB≅∠GED (Given)2.  ∠BAC is the supp. of ∠FAB;  ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3.  ∠BAC≅∠DEC (≅ Supp. Thm.)4.  ∠ACB≅∠DCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6.  AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)7.  △ABC≅△EDC (by ASA Steps 3, 6, 4)<BR>\n" );
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document.write( "B.<BR>\n" );
document.write( "1.  ∠FAB≅∠GED (Given)2.  ∠BAC is the supp. of ∠FAB;  ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3.  ∠BAC≅∠DEC (≅ Supp. Thm.)4.  ∠ACB≅∠DCE (Adj. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6.  AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)7.  △ABC≅△EDC (by ASA Steps 3, 6, 4)<BR>\n" );
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document.write( "C.<BR>\n" );
document.write( "1.  ∠FAB≅∠GED (Given)2.  ∠BAC is the supp. of ∠FAB;  ∠DEC is thesupp. of ∠DEG (Def. of Supp. ∠s)3.  BC¯¯¯¯¯≅CD¯¯¯¯¯ (≅ Supp. Thm.)4.  ∠ACB≅∠BCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6.  BC¯¯¯¯¯≅CD¯¯¯¯¯ (Def. of mdpt.)7.  △ABC≅△EDC (by SAS Steps 3, 1, 6)<BR>\n" );
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document.write( "D.<BR>\n" );
document.write( "1.  ∠FAB≅∠GED (Given)2.  ∠BAC is the supp. of ∠FAB;  ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3.  BC¯¯¯¯¯≅CD¯¯¯¯¯ (≅ Supp. Thm.)4.  ∠ACB≅∠BCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6.  BC¯¯¯¯¯≅CD¯¯¯¯¯ (Def. of mdpt.)7.  △ABC≅△EDC (by SAS Steps 3, 1, 6)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851316 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=851316\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze each option to determine which one correctly proves that △ABC≅△EDC.\r<BR>\n" );
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document.write( "**Understanding the Diagram and Given Information**\r<BR>\n" );
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document.write( "* We are given that ∠FAB≅∠GED.<BR>\n" );
document.write( "* C is the midpoint of AE¯¯¯¯¯. This means AC¯¯¯¯¯≅EC¯¯¯¯¯.<BR>\n" );
document.write( "* We need to use the given information to prove that △ABC≅△EDC.\r<BR>\n" );
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document.write( "**Analyzing Each Option**\r<BR>\n" );
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document.write( "* **A.**<BR>\n" );
document.write( "    * 1. ∠FAB≅∠GED (Given)<BR>\n" );
document.write( "    * 2. ∠BAC is the supp. of ∠FAB; ∠DEC is the supp. of ∠GED (Def. of Supp. ∠s)<BR>\n" );
document.write( "    * 3. ∠BAC≅∠DEC (≅ Supp. Thm.)<BR>\n" );
document.write( "    * 4. ∠ACB≅∠DCE (Vert. ∠s Thm.)<BR>\n" );
document.write( "    * 5. C is the midpoint of AE¯¯¯¯¯ (Given)<BR>\n" );
document.write( "    * 6. AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)<BR>\n" );
document.write( "    * 7. △ABC≅△EDC (by ASA Steps 3, 6, 4)<BR>\n" );
document.write( "    * This option correctly uses the Angle-Side-Angle (ASA) congruence theorem.\r<BR>\n" );
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document.write( "* **B.**<BR>\n" );
document.write( "    * The only difference is that step 4 states ∠ACB≅∠DCE (Adj. ∠s Thm.). Adjacent angles are not necessarily congruent. Vertical angles are. Therefore, this option has an error.\r<BR>\n" );
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document.write( "* **C.**<BR>\n" );
document.write( "    * This option introduces BC¯¯¯¯¯≅CD¯¯¯¯¯ in steps 3 and 6, which is not supported by the given information. Also it tries to use SAS with angles that are not in the correct order.<BR>\n" );
document.write( "    * Also step 4 says ∠ACB≅∠BCE. Which is incorrect. ∠ACB and ∠DCE are vertical angles.\r<BR>\n" );
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document.write( "* **D.**<BR>\n" );
document.write( "    * This option has the same errors as option C.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Option A correctly uses the given information and the ASA congruence theorem to prove that △ABC≅△EDC.\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "The correct answer is A.<BR>\n" );
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