REVIEWER FOR MODERN GEOMETRY
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The axiomatic System
Eratosthenes calculated the area if the Earth
Consistency- there do not exist in the system
any two axioms, any axiom or theorem or any
two theorems of the form “p” and “ not p”. if
there exists a model for a set of axioms, the set
id consistent.
Independence- a statement is said to be
independent in a set of statements if it is
impossible to derive it from other members of
the set. When the statement being tested is
replaced by its denial, there exists a model for
the new set, then the statement being tested is
independent.
Completeness- If it is possible to add such a
statement, the system is complete. If a system
is categorical, then it is complete.
Euclidean Geometry
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All right angles are equal to one
another.
If a transversal falls on two lines in such
a way that the interior angles on one
side of the transversal are less than two
right angles, then the lines meet on that
side on which the angles are less than
two right angles.
Proofs on Euclid’s 5th Postulate and
Playfair’s version
Euclid 5th Postulate states that, if a straight
line (transversal) falling on two straight
lines make the interior angles on the same
side less than two right angles, the two
straight lines produced indefinitely meet on
the side on which the angles are less than
two right angles.
That is, if 𝛼 + 𝛽 < 180°,then the line will
meet exactly once at the side where the
sum of the angles are less than two right
angles (180°).
Axioms
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Things that are equal to the same thing
are also equal to one another.
If equals are added to equals, then the
whole are equal.
If equals are subtracted from equals,
then the remainders are equal.
Things that coincide with one another
are equal to one another.
The whole is bigger than the part.
Postulates
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A straight line can be drawn from any
point to any point.
A finite straight line can be produced
continuously in a straight line.
A circle may be described with any
point as center and any distance as
radius.
In this case, the lines met somewhere to the
right. It is also possible that the line will
meet to the left provided it satisfies the
given conditions.
Playfair’s Postulate states that, through a
given point not on a given line, there can be
drawn only one line parallel to the given
line.
When proving, assume the other statement
to be true and prove the other.
In this case, we assume Euclid’s 5th
postulate and prove Playfair’s Postulate.
In this case, we will consider the structure
below:
REVIEWER FOR MODERN GEOMETRY
Supposed, line l is the given line and point p
as the given point.
These two lines are parallel if and only If the
sum of the alternating angles (𝜷′ + 𝜶 =
𝟏𝟖𝟎° 𝒂𝒏𝒅 𝜶′ + 𝜷 = 𝟏𝟖𝟎°) are 180°. If
𝛼 𝑎𝑛𝑑 𝛽 = 90°, then 𝛼′ 𝑎𝑛𝑑 𝛽′ =
90°(Euclid’s 27th Proposition).
On the other hand, if 𝜶 + 𝜷 < 𝟏𝟖𝟎°, then
the lines meet to the right and if 𝛼 + 𝛽 >
180°, then the lines meet somewhere to the
left. Thus, line m through point P parallel to
line l if and only if 𝜶 + 𝜷 = 𝟏𝟖𝟎°.
Playfair’s Postulate is true and prove
Euclid’s 5th Postulate.