document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210461>Question 1210461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram, AD bisects BAC.\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "By adding one more condition, we can prove that triangles ABD and ACD are congruent.  Which one of the following could be that condition?  Select all that apply.\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "(a) BC = AD\r<BR>\n" );
document.write( "\n" );
document.write( "(b) AB = PQ\r<BR>\n" );
document.write( "\n" );
document.write( "(c) angle ABC = angle BPQ\r<BR>\n" );
document.write( "\n" );
document.write( "(d) angle BCD + angle PAQ = 90 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852735 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The goal is to find an additional condition that proves $\triangle ABD \cong \triangle ACD$.\r<BR>\n" );
document.write( "\n" );
document.write( "## 📐 Congruence Criteria\r<BR>\n" );
document.write( "\n" );
document.write( "We are given two facts about $\triangle ABD$ and $\triangle ACD$:\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Shared Side:** $AD$ is common to both triangles, so **$AD = AD$ (Side)**.<BR>\n" );
document.write( "2.  **Angle Bisector:** $AD$ bisects $\angle BAC$, which means the angle at the vertex $A$ is divided into two equal angles. Therefore, **$\angle BAD = \angle CAD$ (Angle)**.\r<BR>\n" );
document.write( "\n" );
document.write( "So far, we have one **Side** and one **Angle** ($S$ and $A$). To prove congruence, we need one of the following criteria:\r<BR>\n" );
document.write( "\n" );
document.write( "* **SAS** (Side-Angle-Side): We need the other side adjacent to the known angle: $AB = AC$.<BR>\n" );
document.write( "* **AAS** (Angle-Angle-Side): We need another angle and a non-included side: $\angle ABD = \angle ACD$ (and $AD=AD$) or $\angle ADB = \angle ADC$ (and $AD=AD$).<BR>\n" );
document.write( "* **ASA** (Angle-Side-Angle): We need another angle, and the side must be the included side: $\angle ADB = \angle ADC$.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 🔎 Evaluating the Options\r<BR>\n" );
document.write( "\n" );
document.write( "The options introduce segments and angles from other potential figures ($PQ$, $BPQ$, $PAQ$), which are not necessarily part of $\triangle ABD$ or $\triangle ACD$. We must assume that the non-$A, B, C, D$ elements in the options are meant to relate to the required conditions through substitution or equivalence.\r<BR>\n" );
document.write( "\n" );
document.write( "### (a) $BC = AD$\r<BR>\n" );
document.write( "\n" );
document.write( "**False.** This compares the side $BC$ of the entire triangle to the segment $AD$. This is not one of the required congruence conditions.\r<BR>\n" );
document.write( "\n" );
document.write( "### (b) $AB = PQ$\r<BR>\n" );
document.write( "\n" );
document.write( "**True (Potentially).** The required condition for **SAS** congruence is **$AB = AC$**.<BR>\n" );
document.write( "If the condition given is **$AB = PQ$**, and we assume the intended required condition **$AB = AC$** can be satisfied by assuming $PQ$ is equal to $AC$ (i.e., $PQ = AC$), then $AB = AC$.<BR>\n" );
document.write( "More simply, if we assume the statement means $AB = AC$, then **SAS** is satisfied: $\mathbf{AB = AC}$ (Side), $\mathbf{\angle BAD = \angle CAD}$ (Angle), $\mathbf{AD = AD}$ (Side).\r<BR>\n" );
document.write( "\n" );
document.write( "### (c) $\angle ABC = \angle BPQ$\r<BR>\n" );
document.write( "\n" );
document.write( "**True (Potentially).** The required condition for **AAS** congruence is $\angle ABD = \angle ACD$, which is the same as **$\angle ABC = \angle ACB$**.<BR>\n" );
document.write( "If the condition given is $\mathbf{\angle ABC = \angle ACB}$, then **AAS** is satisfied: $\mathbf{\angle ABD = \angle ACD}$ (Angle), $\mathbf{\angle BAD = \angle CAD}$ (Angle), $\mathbf{AD = AD}$ (Non-included Side).<BR>\n" );
document.write( "Assuming the statement intends to mean $\angle ABC = \angle ACB$.\r<BR>\n" );
document.write( "\n" );
document.write( "### (d) $\angle BCD + \angle PAQ = 90^\circ$\r<BR>\n" );
document.write( "\n" );
document.write( "**False.** This introduces a numerical constraint on two angles and does not provide an equality of sides or angles between $\triangle ABD$ and $\triangle ACD$.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## ⭐️ Conclusion (Standard Geometry Test)\r<BR>\n" );
document.write( "\n" );
document.write( "In a typical geometry problem of this nature, the options are testing whether the student knows the necessary congruence criteria, even if the distracting labels ($PQ$, $BPQ$) are used.\r<BR>\n" );
document.write( "\n" );
document.write( "The two easiest ways to prove congruence, given the existing $\mathbf{S}$ and $\mathbf{A}$, are by establishing the **SAS** or **AAS** criteria.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **SAS requires $AB = AC$**<BR>\n" );
document.write( "2.  **AAS requires $\angle ABC = \angle ACB$**\r<BR>\n" );
document.write( "\n" );
document.write( "Since $\triangle ABD$ and $\triangle ACD$ are the focus, we must select the options that are equivalent to $AB=AC$ or $\angle ABC = \angle ACB$.\r<BR>\n" );
document.write( "\n" );
document.write( "* (b) is equivalent to $AB = AC$.<BR>\n" );
document.write( "* (c) is equivalent to $\angle ABC = \angle ACB$.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, both (b) and (c) could be the necessary additional condition, assuming the external angles/segments ($PQ, BPQ$) are equivalent to the required internal ones ($AC, ACB$).\r<BR>\n" );
document.write( "\n" );
document.write( "The statements that could be that condition are **(b)** and **(c)**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );