document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170173>Question 1170173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have two right triangles that are connected at point x. On the right, there is triangle one which has point v at the top, w at the corner, and x which connects to triangle two which has point z at the bottom, connected to y which connects to x.\r<BR>\n" );
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document.write( "I am given that line WV is perpendicular to line WY. I am also given that line ZY is perpendicular to line WY, as well as line WV is congruent to ZY. I am supposed to prove that the triangles are congruent and X is the midpoint of line VZ.  \r<BR>\n" );
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document.write( "I do have a 4th statement that I attempted to do. That is that angle W is congruent to angle Y because of the def. of perpendicularity. I also have about half of the fifth statement which is that angle 1 and angle 2 are congruent. I don't know how to prove this with reason. I need help.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851246 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=851246\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this geometry proof step-by-step.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Triangle VWX and Triangle YXZ<BR>\n" );
document.write( "* WV ⊥ WY<BR>\n" );
document.write( "* ZY ⊥ WY<BR>\n" );
document.write( "* WV ≅ ZY\r<BR>\n" );
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document.write( "**To Prove:**\r<BR>\n" );
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document.write( "1.  Triangle VWX ≅ Triangle YXZ<BR>\n" );
document.write( "2.  X is the midpoint of VZ\r<BR>\n" );
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document.write( "**Proof:**\r<BR>\n" );
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document.write( "Here's the breakdown of the proof with reasons:\r<BR>\n" );
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document.write( "1.  **Statement:** WV ⊥ WY<BR>\n" );
document.write( "    **Reason:** Given\r<BR>\n" );
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document.write( "2.  **Statement:** ZY ⊥ WY<BR>\n" );
document.write( "    **Reason:** Given\r<BR>\n" );
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document.write( "3.  **Statement:** ∠W and ∠Y are right angles.<BR>\n" );
document.write( "    **Reason:** Definition of perpendicular lines.\r<BR>\n" );
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document.write( "4.  **Statement:** ∠W ≅ ∠Y<BR>\n" );
document.write( "    **Reason:** All right angles are congruent.\r<BR>\n" );
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document.write( "5.  **Statement:** WV ≅ ZY<BR>\n" );
document.write( "    **Reason:** Given\r<BR>\n" );
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document.write( "6.  **Statement:** ∠WX V ≅ ∠YX Z<BR>\n" );
document.write( "    **Reason:** Vertical angles are congruent.\r<BR>\n" );
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document.write( "7.  **Statement:** Triangle VWX ≅ Triangle YXZ<BR>\n" );
document.write( "    **Reason:** Angle-Angle-Side (AAS) Congruence Theorem. (We have ∠W ≅ ∠Y, ∠WX V ≅ ∠YX Z, and WV ≅ ZY)\r<BR>\n" );
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document.write( "8.  **Statement:** VX ≅ XZ<BR>\n" );
document.write( "    **Reason:** Corresponding Parts of Congruent Triangles are Congruent (CPCTC)\r<BR>\n" );
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document.write( "9.  **Statement:** X is the midpoint of VZ<BR>\n" );
document.write( "    **Reason:** Definition of a midpoint (a point that divides a segment into two congruent segments).\r<BR>\n" );
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document.write( "**Explanation of Step 6:**\r<BR>\n" );
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document.write( "* When two lines intersect, the angles opposite each other at the intersection point are called vertical angles.<BR>\n" );
document.write( "* Vertical angles are always congruent.<BR>\n" );
document.write( "* In this case, line VX intersects line ZY at point X, creating ∠WX V and ∠YX Z, which are vertical angles.\r<BR>\n" );
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document.write( "**In summary, here is the proof in a table format:**\r<BR>\n" );
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document.write( "| Statement | Reason |<BR>\n" );
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document.write( "| 1. WV ⊥ WY | Given |<BR>\n" );
document.write( "| 2. ZY ⊥ WY | Given |<BR>\n" );
document.write( "| 3. ∠W and ∠Y are right angles | Definition of perpendicular lines |<BR>\n" );
document.write( "| 4. ∠W ≅ ∠Y | All right angles are congruent |<BR>\n" );
document.write( "| 5. WV ≅ ZY | Given |<BR>\n" );
document.write( "| 6. ∠WX V ≅ ∠YX Z | Vertical angles are congruent |<BR>\n" );
document.write( "| 7. Triangle VWX ≅ Triangle YXZ | AAS Congruence Theorem |<BR>\n" );
document.write( "| 8. VX ≅ XZ | CPCTC |<BR>\n" );
document.write( "| 9. X is the midpoint of VZ | Definition of a midpoint |<BR>\n" );
document.write( " </SPAN>\n" );
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