Formal & Informal Geometry
Formal & Informal Geometry MATH 3305
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This 6 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3305 at University of Houston taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/208419/math-3305-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Math 3305 Euclidean Axioms Introduction to Axiomatic Systems The SMSG Axioms for Euclidean Geometry Points Lines Planes and Distance Axioms l 2 3 Coordinates Axioms 3 4 5 6 and 7 Polygons Axiom 8 Convexity and Separation Issues Axioms 9 and 10 Angles Axioms ll 7 l4 Congruent Triangles Axiom 15 Parallel Lines Axiom 16 Area and Congruent Triangles Axioms l7 7 20 Volumes and Solids Axiom 21 and 22 Answers to exercises Glossary Appendix A Why Does the Geometric Coordinate Formula Work Appendix B Theorem List Introduction to Axiomatic Systems In studying any geometry it is important to note the axiomatic framework of the geometry and keep it in mind Often students are so challenged by the details that they forget that there is a structure to geometry Each geometry has a framework called its axiomatic system An outline of a typical axiomatic system is below Any axiomatic system has four parts undefmed terms axioms also called postulates de nitions theorems The undefmed terms are a short list of nouns and relationships These terms may be visualized but cannot be de ned Any attempt at a de nition ends up circling around the terms and using one to de ne the other These are the basic building blocks of the geometry It is usually a good idea to have a mental image of the undefmed terms 7 a visualization of the objects and how they relate Axioms or postulates are a list of rules that de ne the basic relationships among the undefmed terms and make clear the fundamentals facts about a system Axioms are always true for the system No deviation from the facts they state is permitted in working with the system De nitions and theorems build on the axioms and undefined terms clarifying relationships and auxiliary facts We will be using with slight modification the set of undefmed terms and axioms developed by The School Mathematics Study Group during the 1960 s for this module This list of axioms is not as brief as one that would be used by graduate students in a mathematics program nor as long as some of those systems in use in middle school textbooks One defmite advantage to the SMSG list is that it is public domain by design We will be using the Cartesian coordinate plane as our visualization of the undefmed terms of Euclidean geometry Once we have spent time learning the axioms some definitions and a few theorems we will move to the second module on Euclidean Topics and look at geometric shapes and proofs that require using the axioms de nitions and theorems in concert We are studying Euclidean Geometry in this module and it is assumed that you know quite a bit already We will nd more axiomatic systems in the Other Geometries module The SMSG Axioms for Euclidean Geometry Unde ned Terms point line and plane We take as our beginning point the unde ned terms point line and plane Most people visualize a point as a tiny tiny dot Lines are thought of as long seamless concatenations of points and planes are composed of nely interwoven lines smooth endless and at Think of unde ned terms as the basic sounds in a language 7 the sounds that make up our language for the most part have no meaning in themselves but are combined to make words The grammar of our language and a good dictionary are what make the meaning of the sounds This part of language corresponds to the axioms and de nitions that you will nd next in the module From there the facts ights of fancy and contentladen sentences are built 7 these are the theorems and de nition in an axiomatic system The conventions of the Cartesian plane are well suited to assisting in visualizing Euclidean geometry However there are some differences between a geometric approach to points on a line and an algebraic one as we will see in the explanation of Axiom 3 Axioms We will study the axioms in 5 sections The rst eight axioms deal with points lines planes and distance We then look at convexity and separation issues 7these two axioms deal with facts about the relationships among our unde ned objects on a set theoretic basis Axioms 11 through 14 introduce angles measuring and constructing them as well as some fundamental facts about linear pairs With Axiom 15 we begin to look at congruent triangles 7 note that this is so fundamental a notion that it requires its own axiom Axiom l6 introduces parallel lines We then look at area for polygons and congruent triangles axioms l7 7 20 and we nish up with two axioms about solid gures A1 A2 A3 A4 A5 A6 A7 A8 A9 SMSG Postulates for Euclidean Geometry Given any two distinct points there is exactly one line that contains them The Distance Postulate To every pair ofdistinct points there corresponds a unique positive number This number is called the distance between the two points The Ruler Postulate The points ofa line can be placed in a correspondence with the real numbers such that A To every point ofthe line there corresponds exactly one real number B To every real number there corresponds exactly one point ofthe line and C The distance between two distinct points is the absolute value of the difference ofthe corresponding real numbers The Ruler Placement Postulate Given two points P and Q ofa line the coordinate system can be chosen in such a way that the coordinate ofP is zero and the coordinate on is positive A Every plane contains at least three non collinear points B Space contains at least four non coplanar points Iftwo points line in a plane then the line containing these points lies in the same plane Any three points lie in at least one plane and any three non collinear points lie in exactly one plane If two planes intersect then that intersection is a line The Plane Separation Postulate Given a line and a plane containing it the points ofthe plane that do not lie on the line form two sets such that A each of the sets is convex and B ifP is in one set and Q is in the other then segment PQ intersects the line The Space Separation Postulate The points of space that do not line in a given plane form two sets such that A each ofthe sets is convex and B ifP is in one set and Q is in the other then the segment PQ intersects the plane The Angle Measurement Postulate To every angle there corresponds a real number between 0 and 180 The Angle Construction Postulate Let AB be a ray on the edge ofthe half plane H For every r between 0 and 180 there is exactly one ray AP with P in H such that m A PAB r The Angle Addition Postulate IfD is a point in the interior of BAC then szACszADmADAC The Supplement Postulate If two angles form a linear pair then they are supplementary The SAS Postulate Given an one to one correspondence between two triangles or between a triangle and itself Iftwo sides and the included angle ofthe rst triangle are congruent to the corresponding parts ofthe second triangle then the correspondence is a congruence The Parallel Postulate Through a given external point there is at most one line parallel to a given line To every polygonal region there corresponds a unique positive number called its area Iftwo triangles are congruent then the triangular regions have the same area Suppose that the region R is the union oftwo regions R1 and R2 Ile and R2 intersect at most in a nite number of segments and points then the area ofR is the sum ofthe areas ole and R2 The area ofa rectangle is the product ofthe length ofits base and the length ofits altitude
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