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A Note on Notation
It is very important that you are careful when using notation determined by points. For
distinct, non-antipodal points A and B:
• AB indicates the distance between A and B (a number).
• AB indicates the segment with A and B as endpoints (a set).
−→
• AB indicates the ray with A as an endpoint and B as an interior point (a set).
←→
• AB indicates the line which contains A and B (a set).
I have seen many students use AB for all four things. This has led to many situations where
I have been completely confused as to what was meant. If I have been able to determine
what was meant I generally have made a correction and not penalized the work. However,
this has happened so frequently and caused enough confusion that in the future I will begin
penalizing homework solely for the incorrect use of notation.
Similarly, given three points A, B and C:
• (ABC) is the statement that B is in between A and C (a logical statement).
−→
−−→
• ∠ABC is the angle made from rays BA and BC (a set).
• Int ∠ABC is the interior of that angle (a set).
• m∠ABC is the measurement of that angle (a number).
• 4ABC is the triangle with vertices A, B and C (a set).
• Int 4ABC is the interior of that triangle (a set).
• ABC has not been given a meaning.
A Note on What We Have Established
A very common question in a beginning Geometry class is: what can we assume when writing
a proof. On one level, this is a simple to answer. We can say that things are true exactly
when:
• They are assumed in an axiom.
• They have been proven in a theorem that we have proven, or results from statements in
Algebra (since we are assuming the familiar algebraic properties of the real numbers).
• They have been assumed in the hypothesis of a theorem which we are proving, or given
to us in the statement of a problem.
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• They are part of the definition of a statement known to be true, or part of the definition
of an object known to exist.
• We are breaking the problem down into cases, and we are working on an individual
case where the statement is true (though to complete the argument we will eventually
have to deal with all the other cases).
• We are assuming the statement precisely to show that it is impossible (a proof by
contradiction).
If the statement does not fit into these categories, we cannot say that it must be true.
But often this is not of practical help, since we have quite a few results and definitions. It is
also tempting to say more than we can since some things seem very obvious. However, we
cannot say that something must be true simply because we have drawn a picture where it
seems to be true. There are several problems with “proof by picture:”
• Our pictures may not be accurate. For example, it may be that when we draw three
points that they look like they are on a line, but perhaps they are not quite on a line.
For example, if A, B and C are points so that m∠ABC = 179 degrees, then to our
eyes they will look very close to being on a line but in fact no line will go through all
three. There are many “paradoxes” that you can see which use this trick to “prove”
impossible results.
• Our pictures may not cover all relevant situations. For example, if we draw a triangle,
it has to be either acute, right or obtuse. There are some results for each type of
triangle that are not true for the other. For example, if we draw the altitude of an
acute triangle (the line containing a vertex which is perpendicular to the line containing
the opposite side of that vertex), then the altitude will intersect the opposite side of
the triangle. However, in a right triangle one of the sides can be on the altitude, and
in an obtuse triangle the altitude will leave the triangle. If our proof depends on the
altitude intersecting a side of the triangle, it will not work in those cases. Usually when
this happens it means that we need to break the proof up into cases, but if we only
look at one picture we may not realize this.
• Finally, our pictures are necessarily on a flat surface and therefore follow the properties
of Euclidean geometry. But while many things are true in both Euclidean geometry
as well as spherical geometry (in particular almost every result from chapters 3 and
4), they are different in other ways. If you are not explicitly told to assume that you
are working on one geometry, your proofs should work in every valid geometry, and so
you should everything you say should be simultaneously true in every valid geometry
(Euclidean, Spherical, Hyperbolic).
Let’s expand on the last point: what sorts of things are true only in Euclidean geometry,
meaning that we should not make use of them until we are explicitly working in Euclidean
geometry? Some of the biggest are as follows:
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• In Euclidean geometry, given a line ` and a point P not on `, there is exactly one line
which contains P and which is parallel to ` (that it is, it does not intersect `). In
Hyperbolic geometry there is more than one such line, and in Spherical geometry lines
are never parallel.
• In Euclidean geometry the sum of angle measures in a single triangle will be exactly 180
degrees, while in Hyperbolic geometry the sum will be less and in Spherical geometry
the sum will be more.
• In Euclidean geometry it is possible for triangles to be similar without being congruent:
that is, they will have all the same angles and sides with proportional lengths. In
Hyperbolic geometry and Spherical geometry this is impossible, since there if two
triangles have the same angles then they must have all the same side lengths. (This
comes from the fact that in Hyperbolic and Spherical geometry the angle sum of a
triangle determines its area, while in Euclidean geometry all triangles have the same
angle sum so the angle sum does not determine area).
• In Euclidean geometry the sides length of a right triangle satisfy the equation a2 + b2 =
c2 . In Spherical and Hyperbolic geometry the two sides of the equation are never equal.
• The standard methods of trigonometry do not work in Spherical and Hyperbolic
trigonometry. There are methods of Spherical and Hyperbolic trigonometry (which
are detailed in Chapter 6 of our book), but they use different relations of trigonometric functions (and Hyperbolic trigonometry makes use of the hyperbolic trigonometric
functions).
What this means is that until we decide to work in one of these three geometries (which
we will in Chapter 5), it is almost certain that any statement involving parallel lines, angle
sums in triangles, similar triangles, the Pythagorean theorem, or trigonometry will not be
something which we can prove. Therefore if your proof depends on one of those things, it is
most likely invalid.
Most commonly you will be tempted to use an angle sum argument to show that two angles
are congruent. If they seem to be congruent and you need that for your proof, then there
is probably some other way to prove that they are congruent which would be valid in all
three geometries (for example by showing that they are a vertical pair, that they are base
angles of an isosceles triangle, or by showing that they are corresponding angles in a pair of
congruent triangles).
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