document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210462>Question 1210462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Triangles RST and XYZ are congruent.  Point P lies on ST, and point W lies on XY.  Select all the true statements.\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "(a) If RST is isosceles, then XYZ is isosceles.\r<BR>\n" );
document.write( "\n" );
document.write( "(b) If ST = 6, then WY = 6\r<BR>\n" );
document.write( "\n" );
document.write( "(c) If RP \perp ST, then WY \perp YZ\r<BR>\n" );
document.write( "\n" );
document.write( "(d) If RP bisects SRT, then XYZ is right. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852734 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</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=852734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's an analysis of the statements based on the fact that $\triangle RST \cong \triangle XYZ$ (congruent triangles).\r<BR>\n" );
document.write( "\n" );
document.write( "Since the triangles are congruent, their **corresponding vertices, sides, and angles** are equal. The order of the letters defines the correspondence:<BR>\n" );
document.write( "* $R \leftrightarrow X$<BR>\n" );
document.write( "* $S \leftrightarrow Y$<BR>\n" );
document.write( "* $T \leftrightarrow Z$\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the corresponding parts are:<BR>\n" );
document.write( "* **Sides:** $RS = XY$, $ST = YZ$, $RT = XZ$<BR>\n" );
document.write( "* **Angles:** $\angle RST = \angle XYZ$, $\angle STR = \angle YZX$, $\angle TRS = \angle ZXY$\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's evaluate each statement:\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 🧐 Evaluating the Statements\r<BR>\n" );
document.write( "\n" );
document.write( "### (a) If RST is isosceles, then XYZ is isosceles.\r<BR>\n" );
document.write( "\n" );
document.write( "**True.**<BR>\n" );
document.write( "If $\triangle RST$ is isosceles, two of its sides are equal (e.g., $RS = RT$). Since the triangles are congruent, the corresponding sides are also equal ($RS = XY$ and $RT = XZ$). By substitution, $XY = XZ$. If two sides of $\triangle XYZ$ are equal, it is also **isosceles**.\r<BR>\n" );
document.write( "\n" );
document.write( "### (b) If $ST = 6$, then $WY = 6$\r<BR>\n" );
document.write( "\n" );
document.write( "**False.**<BR>\n" );
document.write( "The corresponding side to $ST$ is **$YZ$**, so $YZ = 6$.<BR>\n" );
document.write( "$W$ lies on $XY$, and $Y$ is a vertex. The length $WY$ is just a segment of the side $XY$. The length of $WY$ is generally not equal to $ST$.\r<BR>\n" );
document.write( "\n" );
document.write( "### (c) If $RP \perp ST$, then $WY \perp YZ$\r<BR>\n" );
document.write( "\n" );
document.write( "**False.**<BR>\n" );
document.write( "* The line segment $RP$ is perpendicular to the side $ST$. This means $RP$ is an **altitude** of $\triangle RST$.<BR>\n" );
document.write( "* The corresponding segment in $\triangle XYZ$ is $XW$ (since $R \leftrightarrow X$, $P \leftrightarrow W$, and $S \leftrightarrow Y$, $T \leftrightarrow Z$).<BR>\n" );
document.write( "* The corresponding side to $ST$ is $YZ$.<BR>\n" );
document.write( "* Therefore, the congruent statement is: **If $RP \perp ST$, then $XW \perp YZ$** (meaning $XW$ is an altitude of $\triangle XYZ$).<BR>\n" );
document.write( "* The statement asks if $WY \perp YZ$. $WY$ is a segment of the side $XY$ and is generally **not** perpendicular to $YZ$.\r<BR>\n" );
document.write( "\n" );
document.write( "### (d) If $RP$ bisects $\angle SRT$, then $XYZ$ is right.\r<BR>\n" );
document.write( "\n" );
document.write( "**False.**<BR>\n" );
document.write( "* If $RP$ bisects $\angle SRT$, it means $RP$ is an **angle bisector** of $\angle R$.<BR>\n" );
document.write( "* Since $R \leftrightarrow X$, $XW$ is the corresponding angle bisector of $\angle X$ in $\triangle XYZ$.<BR>\n" );
document.write( "* If the angle bisector of a triangle is also the altitude (from statement (c)), the triangle must be **isosceles**. This means $RS = RT$ and $XY = XZ$.<BR>\n" );
document.write( "* $RP$ being an angle bisector does **not** make $\triangle RST$ a right triangle, and thus it does not make $\triangle XYZ$ a right triangle. $\triangle XYZ$ would be isosceles.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "The only true statement is **(a)**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );