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.