document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972033>Question 972033</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that if one pair of opposite sides of a quadrilateral is both congruent and parallel, the quadrilateral is a parallelogram.\r<BR>\n" );
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document.write( "Given: Line BC is parallel to line DA, Line BC is congruent to line DA.<BR>\n" );
document.write( "Prove: ABCD is a parallelogram.\r<BR>\n" );
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document.write( "ABCD looks like a parallelogram with line BC and AD opposite to each other. It also shows that BC AD are parallel and congruent.\r<BR>\n" );
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document.write( "PLEASE SHOW EACH STEP FOR ME. I WOULD GREATLY APPRECIATE IT :)\r<BR>\n" );
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document.write( "When showing the answer, if possible, please have the answer look like this:\r<BR>\n" );
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document.write( "Statements............... Reasons<BR>\n" );
document.write( "1...............................1.<BR>\n" );
document.write( "2...............................2.<BR>\n" );
document.write( "3...............................3.<BR>\n" );
document.write( "4...............................4.<BR>\n" );
document.write( "etc............................etc. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #594496 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=594496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I have come to the conclusion that all of geometry boils down to triangles and their properties.<BR>\n" );
document.write( "You just have to find the right triangles, drawing extra lines/segments as needed.<BR>\n" );
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document.write( "In this case, you would need to use congruent triangles (and their properties and theorems),<BR>\n" );
document.write( "and the facts that<BR>\n" );
document.write( "when two lines are cut by a transversal, the alternate interior angles are congruent,<BR>\n" );
document.write( "and that<BR>\n" );
document.write( "if two lines are cut by a transversal, and the alternate interior angles are congruent, then the lines are parallel.<BR>\n" );
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document.write( "If you draw the lines that contain the sides,<BR>\n" );
document.write( "and the transversal line that contains diagonal AC,<BR>\n" );
document.write( "you find that the transversal line that intersects parallel lines BC and AD, and splits the quadrilateral into triangles BCA and DAC.<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-3%2C33%2C-2%2C22%2C%0D%0Aline%2811%2C-0.7%2C11%2C0.7%29%2Cline%2816%2C19.3%2C16%2C20.7%29%2C%0D%0Aarrow%282%2C0%2C14%2C0%29%2Carrow%287%2C20%2C19%2C20%29%2C%0D%0Aline%28-3%2C0%2C33%2C0%29%2Cline%28-3%2C20%2C33%2C20%29%2C%0D%0Aline%28-1%2C-4%2C6%2C24%29%2Cline%2824%2C-4%2C31%2C24%29%2C%0D%0Ablue%28arrow%28-3%2C-2%2C39%2C26%29%29%2C%0D%0Alocate%28-1.2%2C1.7%2CA%29%2Clocate%2825.2%2C1.7%2CD%29%2C%0D%0Alocate%283.5%2C20%2CB%29%2Clocate%2830%2C20%2CC%29%2C%0D%0Agreen%28triangle%281%2C0.4%2C24.6%2C0.4%2C29.6%2C19.3%29%29%2C%0D%0Ared%28triangle%280.2%2C0.4%2C5.2%2C19.8%2C29.5%2C19.8%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > You have to prove those triangles are congruent.<BR>\n" );
document.write( "Then you use the theorem (usually abbreviated CPCTC) that says that<BR>\n" );
document.write( "Corresponding Parts of Congruent Triangles are Congruent,<BR>\n" );
document.write( "to prove that the alternate interior angles the transversal forms with the other two sides are congruent.<BR>\n" );
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document.write( "You can find that proof (in the boring Statement?reason table format in http://www.regentsprep.org/regents/math/geometry/gp9/lparallelogram.htm. <BR>\n" );
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document.write( "Statements 1 and 2 (or both lumped together as stament 1) could be <BR>\n" );
document.write( "Line BC is parallel to line DA, and Segment BC is congruent to Segment DA.<BR>\n" );
document.write( "The reason for that is written as \"Given\", because it is information given to you as part of the problem.<BR>\n" );
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document.write( "Next Statement:<BR>\n" );
document.write( "The pair of alternate interior angles BCA and DAC area congruent.<BR>\n" );
document.write( "Reason: Because AC is a transversal line intersecting parallel lines BC and AD.\r<BR>\n" );
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document.write( "(A silly extra statement may be required; the diagonal as a side of one triangle is congruent to the diagonal as a side of the other triangle, for the reason that it is the same segment, and any segment is congruent with itself).<BR>\n" );
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document.write( "From there, you prove that the two triangles formed as the diagonal splits the quadrilateral are congruent.<BR>\n" );
document.write( "The reason is that they have a pair of congruent sides flanking a congruent angle (SAS congruency).<BR>\n" );
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document.write( "Next statement is that angles ACD and BAC are congruent.<BR>\n" );
document.write( "Since the triangles are congruent, their corresponding parts are congruent (you could write as reason \"by CPCTC\").<BR>\n" );
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document.write( "Next statement is that the other two sides (AB and CD) are parallel.<BR>\n" );
document.write( "The reason is that if the alternate interior angles formed by a transversal (like AC) with two lines {like lines AB and CD) are congruent,<BR>\n" );
document.write( "then those two lines are parallel.<BR>\n" );
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document.write( "Next statement is that ABCD is a parallelogram.<BR>\n" );
document.write( "The reason is it has two sets of parallel sides (AD||CD and BC||AD),<BR>\n" );
document.write( "and a quadrilateral with two sets of parallel sides is by definition a parallelogram. </SPAN>\n" );
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