Introduction to Neutral Geometry

1. Neutral
Geometry
Part 2

2. Introduction
Euclidean Geometry cannot
describe all physical aspect
such as curved space
Euclid proposed 5
postulates (axioms) which
are considered true
Among the 5
postulates, the fifth is
the most criticized.
Euclidean Geometry
tekhnologic
Bless
Ronald
SPIN

3. Introduction
Introduction The Most Puzzling
Postulate of Euclid
The FIFTH POSTULATE
Parallel Postulate
< 90°
< 90°
meet
Hyperbol
ic

4. Introduction
Introduction
-is used to answer the question
what can be proved without
using a parallel axiom
If you add E5P (or something
equivalent) to Neutral
Geometry, then you get
Euclidean Geometry. But if you
add the negation instead, you
get Hyperbolic Geometry.
So Neutral Geometry
gives theorems that are
common to both of
these important
geometries
Alternate Interior Angle Theorem
If alt. int angle are congruent then lines are
parallel.
Exterior Angle
Theorem
Exterior angle is
greater than remote
interior
Measure of angles
and
segments theorem
Saccheri-Legendre Theorem
The sum of the interior angles of a triangle
is at most 180
The sum of the measures of two angles of a
triangle is less than 180
Equivalence
of parallel
postulates:
are all
equivalent.
Hilbert parallelism
axiom
Euclid V
Converse to Alt. Int.
Angle theo
Sum of int ang of
triangle 180
David Hilbert(1862-
1943)- writes 14 axioms…
Divided into 4 Axioms:
Incidence,
congruence, and continuity
betweenness,

5. Segments
and
Angles
O F
A N G L E S A N D
S E M E N T S
M E A S U R E S
Segments
and
Angles
Given a segment OI, called
the unit segment. Then
there is a unique way of
assigning a
length AB to each segment
AB such that the following
properties hold:
O I
A
AB
B
A B

6. Segments
and
Angles
Segments
and
Angles
(a)AB is a positive real number
and OI = 1.
(b) AB = CD if and only AB @ CD
(c) A*B*C if and only if
AC = AB + BC .
(d) AB < CD if and only if
AB < CD
(e) For every positive real
number x, there exists a
segment AB such that AB= x
O I
1
A B C D
x
A B

7. Segments
and
Angles
Segments
and
Angles
A B
Let 𝐵1be the midpoint of AB
𝐵1
Let 𝐵2be the midpoint of AB1
𝐵2 𝐵2
In general, 𝐵𝑘+1be the
midpoint of AB𝑘
Intuitively, 𝐴𝐵𝑘 should be the
1/2𝑘
There is a point between any
two separated partitions of a
line.
Methods of Assigning Length
AB will be determined as an
infinite decimal

8. Segments
and
Angles
Segments
and
Angles
(a) A° is a real number such
that 0°<  A°<180°
(b) A° = 90° if and only if
A is a right angle
(c) A°= B° if and only
if A @ B .
A
A
A
B

9. Segments
and
Angles
Segments
and
Angles
(d) If AC is interior to  DAB ,
then  DAB°=  DAC°+  CAB°
(e) For every real number x
between 0 and 180, there
exists an angle  A such that
 A°= x°
(f) If  B is
supplementary to  A,
then  A°+  B°= 180°
(g)  A°>  B° if and only
if  A >  B.
A
C
D
B
A
x°
B
A
D

10. Segments
and
Angles
Segments
and
Angles
There is a unique point D such
that A * B * D and BD ≅ BC
(Axiom C-1 applied to the ray
opposite to BA)
Then ∠BCD ≅ ∠ BDC
(Proposition 3.10: base angles
of an isosceles triangle)
C O R O L L A R Y
If A, B, and C are
three noncollinear points,
then AC < AB + BC
Proof:
A B
C
D
AD =AB+ BD (Theorem 4.3(9))
and BD = BC
(Step 1in Theorem 4.3 (8));
substituting gives AD =AB+ BC.
CB is between CA and CD
(Proposition 3.7); hence
∠ ACD < ∠ BCD (by definition)
AD > AC
Hence, AB + BC > AC
(Theorem 4.3(10))

11. THEOREM
S
EGENDRE
L
ACCHERI-
S-L
Theorem
S-L
Theorem
The sum of the
degree measures of
the three angles in
any triangle is less
than or equal to
180°

12. Saccheri-Legendre
Theorem
Lemma 1
S-L
Theorem
S-L
Theorem
A B
C
D
Given ⊿ABC, m∠CAB+m∠ABC < 180
We know that m∠ABC+m∠CBD=180
m∠CBD > m∠CAB
m∠ABC+m∠CAB < 180

13. Saccheri-Legendre
Theorem
Lemma 2
S-L
Theorem
S-L
Theorem
B C
A
E
Given ⊿ABC, with E on BC, then
σ(⊿ABE)+σ(⊿ECA)= σ(⊿ABC) + 180°
𝜶𝟏 𝜶𝟐
𝜹𝟐
𝜹𝟏
𝜷 𝜸
𝛼1 + 𝛽 + 𝛿1 + 𝛼2 + 𝛾 + 𝛿2
𝛽 + 𝛾 + (𝛼1 + 𝛼2) + (𝛿1 + 𝛿2)
σ(⊿ABC) + 180°

14. Saccheri-Legendre
Theorem
Lemma 3
S-L
Theorem
S-L
Theorem
A B
C
Given ⊿ABC, there exists D such that
D is not on the line AB and
σ(⊿ABD)=σ(⊿ABC) and one interior
angle of ⊿ABD is less than or equal
to half of m∠CAB
E
D
σ(⊿AEC)=σ(⊿BED)
σ(⊿ABC)=σ(⊿ABE)+ σ(⊿AEC)-180°
By Lemma 2
σ(⊿ABD)=σ(⊿ABE)+ σ(⊿BED)-180°
σ(⊿ABD)=σ(⊿ABC)
⊿
⊿
⊿ ⊿
∠ ∠

15. Saccheri-Legendre
Theorem
Prove:
If ⊿ABC is any triangle, then
𝛔(⊿ABC)≤180 °
S-L
Theorem
S-L
Theorem
A
B
C
By Contradiction:
Suppose 𝛔(⊿ABC) > 180°
𝛔(⊿ABC) = 180°+ 𝜀° 𝜀°∈ ℜ, 𝜀 > 0
Using Archimedian property
We can choose n such that
2𝑛
𝜀° > 𝑚∠𝐴
∃∆𝜃𝑛 𝑠. 𝑡. 𝜃𝑛 < 𝜀
𝜃1 + 𝜃2 + 𝜃3 … ≤ 𝑚∠𝐴
𝑪𝟏
𝜽𝟏
𝑪𝟐
𝜽𝟐
𝜽𝟐
𝜽𝟏
𝑪𝒏
𝜽𝒏
𝜃1 =
1
2
𝑚∠𝐴
𝜃2 =
1
22 𝑚∠𝐴 … 𝜃𝑛 =
1
2𝑛 𝑚∠𝐴
𝜃2 =
1
2
𝑚∠𝐴1
𝛔(⊿ABC) = 𝛔(⊿AB𝐂𝐧) by Lemma 3
𝛔(⊿AB𝐂𝐧) = 180°+ 𝜀°
< 𝜀, thus 𝜃𝑛< 𝜀

16. Saccheri-Legendre
Theorem
Prove:
If ⊿ABC is any triangle, then
𝛔(⊿ABC)≤180 °
S-L
Theorem
S-L
Theorem
A
B
C Then, 𝜃𝑛 +180 < 𝜀+ 180
𝑪𝟏
𝜽𝟏
𝑪𝟐
𝜽𝟐
𝜽𝟐
𝜽𝟏
𝑪𝒏
𝜽𝒏
𝜃𝑛 =
1
2𝑛 𝑚∠𝐴 < 𝜀, thus 𝜃𝑛< 𝜀
𝜃𝑛 +180 < 𝛔(⊿AB𝐂𝐧)
𝜃𝑛 +180 <𝑚∠𝐶𝑛𝐴𝐵+ 𝑚∠𝐶𝑛𝐵𝐴 +𝜃𝑛
180 < 𝑚∠𝐶𝑛𝐴𝐵+ 𝑚∠𝐶𝑛𝐵𝐴
Which is false by Lemma 1
Hence the assumption
𝛔(⊿ABC) > 180° is false
Therefore,
𝝈(⊿ABC) is not greater than 180°
𝝈(⊿ABC)≤180 °

17. Construct
Let O, P, Q, and R be points on the
neutral plane P
∠ POQ is right, ∠ ROQ is right
and R are on opposite sides of 𝐎𝐐
Q, O and R are collinear
S-L
Theorem
S-L
Theorem
Q
O P
P
R

18. S-L
Theorem Points to Remember
Neutral Geometry or Absolute
geometry is the geometry
derived from the first four
postulates of Euclid, or the
first eleven axioms.
Sacherri-Legendre
Theorem states that the
sum of the degree
measures of the three
angles in any triangle is
less than or equal to 180°
Did you know that?
Sacherri is an Italian Priest
who used the first four
postulates of Euclid in order
to prove the fifth postulate.
Adrien Legendre continued
the work of Saccheri and
Lambert but was still unable
to derive a contradiction in
the acute case thus the
creation of Saccheri-
Legendre Theorem
S-L
Theorem

19. REFERENCES
END
NOTE
The laws of nature are but
a mathematical thoughts
of god
https://www.youtube.com/watch?v=LPET_HhN0VM&t=9s
October,2021 | First Semester 2021-2022
•https://www.youtube.com/watch?v=vs8xgUKpx_o
&list=PLmq7agHYdLul2E76Vdg7LCYAi4IYK5nQz
•https://pediaa.com/what-is-the-difference-
between-postulates-and-theorems/
https://www.youtube.com/watch?v=dEELOzeZJDY
•Geometry Illuminated(an Illustrated Introduction To
Euclidean & Non-euclidean Geometry) by Matthew
Harvey
•Specht, et al: Euclidean Geometry: Exercises and
Answers: Contents
REFERENCES
-EUCLID