document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=762417>Question 762417</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are the basis postulates of the geometry? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #463930 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=463930\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Postulates are statements that are assumed to be true without proof. Postulates serve two purposes - to explain undefined terms, and to serve as a starting point for proving other statements.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=geometry ALIGN=MIDDLE  ALT=\"geometry\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+postulates ALIGN=MIDDLE  ALT=\"+postulates\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "    Point-Line-Plane Postulate<BR>\n" );
document.write( "    A) Unique Line Assumption: Through any two points, there is exactly one line.<BR>\n" );
document.write( "    Note: This doesn't apply to nodes or dots.<BR>\n" );
document.write( "    B) Dimension Assumption: Given a line in a plane, there exists a point in the plane not on that line. Given a plane in space, there exists a line or a point in space not on that plane.<BR>\n" );
document.write( "    C) Number Line Assumption: Every line is a set of points that can be put into a one-to-one correspondence with real numbers, with any point on it corresponding to zero and any other point corresponding to one.<BR>\n" );
document.write( "    Note: This doesn't apply to nodes or dots. This was once called the Ruler Postulate.<BR>\n" );
document.write( "    D) Distance Assumption: On a number line, there is a unique distance between two points.<BR>\n" );
document.write( "    E) If two points lie on a plane, the line containing them also lies on the plane.<BR>\n" );
document.write( "    F) Through three noncolinear points, there is exactly one plane.<BR>\n" );
document.write( "    G) If two different planes have a point in common, then their intersection is a line. \r<BR>\n" );
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document.write( "Euclid's Postulates<BR>\n" );
document.write( "     A) Two points determine a line segment.<BR>\n" );
document.write( "    B) A line segment can be extended indefinitely along a line.<BR>\n" );
document.write( "    C) A circle can be drawn with a center and any radius.<BR>\n" );
document.write( "    D) All right angles are congruent.<BR>\n" );
document.write( "    Note: This part has been proven as a theorem. See below, proof.<BR>\n" );
document.write( "    E) If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180 degrees, then the lines will intersect on that side of the transversal. \r<BR>\n" );
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document.write( "Polygon Inequality Postulates\r<BR>\n" );
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document.write( "    Triangle Inequality Postulate: The sum of the lengths of two sides of any triangle is greater than the length of the third side.<BR>\n" );
document.write( "    Quadrilateral Inequality Postulate: The sum of the lengths of 3 sides of any quadrilateral is greater than the length of the fourth side. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=algebra ALIGN=MIDDLE  ALT=\"algebra\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+postulates ALIGN=MIDDLE  ALT=\"+postulates\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "    Postulates of Equality\r<BR>\n" );
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document.write( "    Reflexive Property of Equality: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+a ALIGN=MIDDLE  ALT=\"a+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Symmetric Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+a ALIGN=MIDDLE  ALT=\"b+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Transitive Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+c ALIGN=MIDDLE  ALT=\"b+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+c ALIGN=MIDDLE  ALT=\"a+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( ".\r<BR>\n" );
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document.write( "Postulates of Equality and Operations\r<BR>\n" );
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document.write( "    Addition Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+c+=+b+%2B+c ALIGN=MIDDLE  ALT=\"a+%2B+c+=+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Multiplication Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2A+c+=+b+%2A+c ALIGN=MIDDLE  ALT=\"a+%2A+c+=+b+%2A+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Substitution Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be substituted for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > in any equation or inequality<BR>\n" );
document.write( "    Subtraction Property of Equality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+-+c+=+b+-+c ALIGN=MIDDLE  ALT=\"a+-+c+=+b+-+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Postulates of Inequality and Operations\r<BR>\n" );
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document.write( "    Addition Property of Inequality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+%3E+b ALIGN=MIDDLE  ALT=\"a+%3C+%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+c+%3C+%3E+b+%2B+c ALIGN=MIDDLE  ALT=\"a+%2B+c+%3C+%3E+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Multiplication Property of Inequality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3E+0 ALIGN=MIDDLE  ALT=\"c+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2A+c+%3C+b+%2A+c ALIGN=MIDDLE  ALT=\"a+%2A+c+%3C+b+%2A+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3C+0 ALIGN=MIDDLE  ALT=\"c+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2A+c+%3E+b+%2A+c ALIGN=MIDDLE  ALT=\"a+%2A+c+%3E+b+%2A+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Equation to Inequality Property: if a and b are positive, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+=+c ALIGN=MIDDLE  ALT=\"a+%2B+b+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3E+a ALIGN=MIDDLE  ALT=\"c+%3E+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3E+b ALIGN=MIDDLE  ALT=\"c+%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    if a and b are negative, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+=+c ALIGN=MIDDLE  ALT=\"a+%2B+b+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3C+a ALIGN=MIDDLE  ALT=\"c+%3C+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3C+b ALIGN=MIDDLE  ALT=\"c+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Subtraction Property of Inequality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+%3E+b ALIGN=MIDDLE  ALT=\"a+%3C+%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+-+c+%3C%3E+b+-+c ALIGN=MIDDLE  ALT=\"a+-+c+%3C%3E+b+-+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Transitive Property of Inequality: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3C+c ALIGN=MIDDLE  ALT=\"b+%3C+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+c ALIGN=MIDDLE  ALT=\"a+%3C+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Postulates of Operation\r<BR>\n" );
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document.write( "    Commutative Property of Addition:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+%2B+b+=+b+%2B+a ALIGN=MIDDLE  ALT=\"+a+%2B+b+=+b+%2B+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Commutative Property of Multiplication: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2A+b+=+b+%2A+a ALIGN=MIDDLE  ALT=\"a+%2A+b+=+b+%2A+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Distributive Property: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2A+%28b+%2B+c%29+=+a+%2A+b+%2B+a+%2A+c ALIGN=MIDDLE  ALT=\"a+%2A+%28b+%2B+c%29+=+a+%2A+b+%2B+a+%2A+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and vv\r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );