document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165628>Question 1165628</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given: AB= DC<BR>\n" );
document.write( "prove: AC= DB </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852956 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=852956\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a great question that tests the definitions and properties of quadrilaterals! However, based on the standard geometric definitions, the statement you are asked to prove, **AC = DB**, is **not generally true** for a kite with the given condition.\r<BR>\n" );
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document.write( "In standard Euclidean geometry:\r<BR>\n" );
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document.write( "1.  A **Kite** is a quadrilateral where two pairs of equal-length sides are adjacent to each other.<BR>\n" );
document.write( "2.  The diagonals of a kite are **perpendicular** ($\overline{AC} \perp \overline{DB}$).<BR>\n" );
document.write( "3.  Only **one** of the diagonals is bisected (the shorter one).\r<BR>\n" );
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document.write( "The statement **AC = DB** means the diagonals are equal, which is a property of a **rectangle** or an **isosceles trapezoid**, but not a general kite.\r<BR>\n" );
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document.write( "Let's assume there might be a property of the specific kite *ABCD* implied by the side equality $\mathbf{AB = DC}$.\r<BR>\n" );
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document.write( "## 📐 Analysis of the Given Information\r<BR>\n" );
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document.write( "### Given: ABCD is a Kite\r<BR>\n" );
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document.write( "The standard side properties of a kite are that two pairs of adjacent sides are equal. There are two possible pairings:\r<BR>\n" );
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document.write( "1.  **Case 1 (Standard):** $AB = AD$ and $CB = CD$.<BR>\n" );
document.write( "2.  **Case 2 (Alternative):** $AB = CB$ and $AD = CD$.\r<BR>\n" );
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document.write( "### Given: AB = DC\r<BR>\n" );
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document.write( "The given condition **$AB = DC$** (adjacent sides equal to non-adjacent sides) is unusual for a general kite, but we must incorporate it.\r<BR>\n" );
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document.write( "If $ABCD$ is a kite, we must combine the kite definition with the given condition:\r<BR>\n" );
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document.write( "* **If we assume Case 1 ($AB = AD$ and $CB = CD$):**<BR>\n" );
document.write( "    * We are given $AB = DC$.<BR>\n" );
document.write( "    * Since $AB=AD$ and $CB=CD$, substituting the given condition means $AD = DC$.<BR>\n" );
document.write( "    * Since $AD = DC$ and $CB = CD$, then $AD=DC=CB$.<BR>\n" );
document.write( "    * If all three sides ($AD, DC, CB$) are equal, the only way $ABCD$ can be a kite is if the fourth side is also equal, making **$ABCD$ a Rhombus** (and thus a Parallelogram).\r<BR>\n" );
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document.write( "* **If $ABCD$ is a Rhombus:** All sides are equal ($AB=BC=CD=DA$). A rhombus is a specific type of kite.\r<BR>\n" );
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document.write( "## 🛑 Proving the Diagonals are Equal\r<BR>\n" );
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document.write( "The statement you want to prove, $\mathbf{AC = DB}$, is only true if the figure is a **rectangle** (or a square).\r<BR>\n" );
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document.write( "If a quadrilateral is *both* a kite (diagonals are perpendicular) *and* a rectangle (diagonals are equal), it must be a **Square**.\r<BR>\n" );
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document.write( "### Conclusion based on the Given Information\r<BR>\n" );
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document.write( "If the given conditions **$AB=DC$** and **$ABCD$ is a kite** force the figure to be a **Rhombus** (as shown above), then:\r<BR>\n" );
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document.write( "* Rhombus properties: Diagonals are perpendicular.<BR>\n" );
document.write( "* **Rhombus properties: Diagonals are NOT necessarily equal.**\r<BR>\n" );
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document.write( "**Conclusion:** The claim $\mathbf{AC = DB}$ cannot be proven from the given information alone.\r<BR>\n" );
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document.write( "If this were intended to be a textbook problem that *is* provable, the quadrilateral must have been defined as something else, such as an **Isosceles Trapezoid** (where $\overline{AC} = \overline{DB}$), or the side condition was meant to imply the figure is a **Square** (which would be an extremely specific case).\r<BR>\n" );
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document.write( "**The statement AC = DB is false for the general conditions AB=DC and ABCD is a kite.** </SPAN>\n" );
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