3.1 - 3.3 - 3.1 3.2 3.3 Triangles Congruence Relations SAS...

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3.1 Triangles, Congruence Relations, SAS Hypothesis 3.2 Taxicab Geometry: Geometry without SAS Congruence 3.3 SAS, ASA, SSS Congruence, and Perpendicular Bisectors Absolute Geometry We are building axioms that will result in one of two choices*: Euclidean Geometry or Hyperbolic Geometry. The collection of axioms through SAS is known as Absolute Geometry. In section 3.3 ½ we get a different set of axioms called Neutral Geometry. These axioms allow choices that will produce Spherical, Hyperbolic, or Euclidean Geometry. We make the choice that creates Spherical Geometry and study it for a bit and then we come back to Absolute Geometry. Homework assignment 3 * in Chapter 6 – we’ve got a way to go 1
3.1 Triangles, Congruence Relations, and SAS Hypothesis This section begins with a very careful definition of a triangle. A triangle is a point set composed of distinct point sets connected in a very specific way. Please read the definition and pay special attention to how much really does need to be said. Figure 3.1 brings some additional vocabulary that you should be familiar with: included side, vertex opposite a side, side opposite an angle, included angle. The author was really saving trees by illustrating these terms rather than defining them in words. Congruence is a property of two sets that is related to equality but not quite the same thing. Set equality means that the elements of one set are exactly the same as the elements of another set (which is to say that there’s one set with two names). Sometimes it’s not a all obvious that there’s only one set and you need to PROVE that there’s only one set with a set containment proof. However, when we say two sets are congruent, we do NOT mean that there’s only one set with two names, we mean that two distinct sets share a property that makes them alike in the sense of that property. The symbol for correspondence, 2245 , is reminiscent of an equality but is not quite an equality symbol. For example: page 122 Two distinct segments are congruent iff their lengths are the same. These two segments are congruent and not equal. They are NOT equal; they are not subsets of one another. What they share is that the number that describes the distance from one endpoint to another is the same number for both sets. We denote this relationship in the following manner: AB DE 2245 . Two distinct angles are congruent iff their measures are equal. Once you move beyond these simple objects, you need the notion of a correspondence to begin talking about congruence. A correspondence is a relationship that identifies the points of one set pair-wise with the points of another set. For triangles we will match vertices and sides between two triangles. There are MANY correspondences possible between two triangles.
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