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Non –Euclidean Geometry
Chapter 4
Alternate Interior Angles Theorem
Thm 4.1: Alterenate Interior Angles
(AIA): If two lines cut by a transversal
have a pair of congruent alternate interior
angles, then the two lines parallel.
Corr 1: Two lines perpendicular to the
same line are parallel.
Corr 2 (Euc I.31): If l is an line and P is
any point not on l, there exists at least one
line m through P parallel to l.
Exterior Angle Theorem
Thm 4.2: Exterior Angle (EA): An exterior angle
of a triangle is greater than either remote interior
angle.
Corr 1: If a triangle has a right or obtuse angle
the other two angles are acute.
P4.1: (SAA) Given AC  DF, A  D and
B  E. Then ABC  DEF.
P4.2: Hypotenuse-Leg Criterion (H-L): Two right
triangles are congruent if the hypotenuse and a leg
of one are congruent respectively to the hypotenuse
and a leg of the other.
P4.3 (Midpoints) Every segment has a unique
midpoint.
P4.4: (Bisectors) Every angle has a unique bisector.
Every segment has a unique perpendicular bisector.
P4.5: In a triangle ABC, the greater angle lies
opposite the greater side and the greater side lies
opposite the greater angle
P4.6: Given ABC and A'B'C', if AB  A'B' and BC 
B'C' then B < B’ iff AC < A'C'.
Thm 4.3: Measurement Theorem
A: There is a unique way of assigning a
degree measure to each angle such
that the following properties hold:
0) (A) is a real number such that 0 < (A) < 180.
1) (A) = 90 iff A is a right angle.
2) (A) = (B) iff A  B.
(more)
Thm 4.3: Measurement Thm (con’d)

3) If AC is interior to DAB,
then (DAB) = (DAC) + (CAB)
4) For every real number x between 0 and 180, there exists an
angle A such that (A) = x .
5) If B is supplementary to A,
then (A) + (B) = 180
6) (A) > (B) iff A > B
(more)
Thm 4.3: Measurement Thm (con’d)
B: Given a segment OI, called a unit segment.
Then there is a unique way of assigning a length
to each segment AB such that the follow
properties hold;
7) AB is a positive real number and OI = 1.
8) AB = CD iff AB  CD.
9) A*B*C iff AC  AB  BC
10) AB < CD iff AB < CD.
11) For every positive real number x, there exists
a segment AB such that AB = x.
Corr 2: to EA Theorem:
The sum of the degree measures of
any two angle of a triangle is less
than 180.
Triangle Inequality:
If A, B, and C are three noncollinear
points, then AC  AB  BC.
Corr : For any Hilbert plane, the
converse to the triangle inequality is
equivalent to the circle-circle
continuity principle. Hence the
converse to the triangle inequality
holds in Euclidean planes.
Euclid 5th Postulate:
If two lines are intersected by a
transversal in such a way that the
sum of the degree measures of the
two interior angles on one side of the
transversal is less than 180, then the
two lines meet on that side of the
transversal .
Theorem 4.4:
Euclid's fifth postulate  Hilbert's
parallel postulate.
(Note: Hilbert/Euclidean Postulate:
For every line l and every point P not
lying on l there is at most one line m
through P such that m is parallel to
l.)
Parallel Equivalencies:
P4.7: Hilbert's parallel postulate  if a line
intersects one of two parallel lines, then it also
intersects the other.
P4.8: Hilbert's parallel postulate  converse to
Theorem 4.1 (alternate interior angles).
P4.9: Hilbert's parallel postulate  if t is a
transversal to l and m, l || m, and t  l, then t  m.
P4.10: Hilbert's parallel postulate  k || l, m k,
and n  l, then either m = n or m || n
P4.11:
In any Hilbert plane, Hilbert’s
Euclidean parallel postulate implies
that for every triangle ABC,
( A) + ( B) + ( C) = 180.
(The angle sum of every triangle is
180 if we assume Hilbert’s Euclidean
parallel postulate.)
Last Stop before Non-Euclidean
Geometry!
Saccheri
And
Lambert
Quadrilaterals
Saccheri Quadrilateral
Def: A quadrilateral
 ABDC is bi-right
if adjacent angles  A and  B are
right angles. AB is called the base; CD
the summit.
Def: A Saccheri quadrilateral is an
isoscoles (CA  DB, called sides) biright quadrilaterial.
See figure:
Lambert Quadrilateral
A quadrilateral with at least three
right angles is called a Lambert
Quadrilateral.
P4.12:
a) (Sac I). The summit angles of a
Saccheri quadrilateral are congruent
to each other.
b) (Sac II). The line joining the
midpoints of the summit and the base
is perpendicular to both the summit
and the base.
P4.13
In any bi-right quadrilateral ABDC,
 C >  D  BD > AC. (The greater
side is opposite the greater summit
angle.)
Corr: 1, 2 & 3 follow
P4.13 Corr: 1
Given any acute angle with vertex V. Let Y
be any point on one side of the angle, let Y’
be any point farther out on the side, i.e.
V*Y*Y’. Let X, X’ be the feet of the
perpendiculars from Y,Y’, respectively, to
the other side of the angle. The Y’X’ > YX.
(In other words: The perpendicular
segments from one side of an acute angle to
the other increase as you move away from
the vertex of the angle.)
P4.13 Corr: 2 & 3
Corr 2: Euclid V implies Aristotle’s
axiom.
Corr 3: (Sac II, Corr I) A side
adjacent to the fourth angle  of a
Lambert quadrilateral is, respectively,
greater than, congruent to, or less
than its opposide side   is acute,
right or obtuse, respectively.
Lambert Quadrilateral
A quadrilateral with at least three
right angles is called a Lambert
Quadrilateral.
P4.13 Corr: 4
Corr 4: (Sac III) the summit of a
Saccheri quadrilateral is,
respectively, greater than, congruent
to, or less than the base  its
summit angle is acute, right or
obtuse, respectively.
Uniformity Theorem:
For any Hilbert plane, if one Saccheri
quadrilateral has acute (respectively,
right, obtuse) summit angles, then so
do all Saccheri quadrilaterals.
Corr: 1,2,3 & 4 follow.
Uniformity Thm:
Corr 1: For any Hilbert plane, if one
Lambert quadrilateral has an acute
(respectively, right, obtuse) fourth
angle, then so do all Lambert
quadrilaterals. Furthermore, the
type of the fourth angle is the same
as the type of the summit angles of
Saccheri quadrilaterals.
Uniformity Theorem:
Corr 2: There exists a rectangle in a
Hilbert plane  the plane is semiEuclidean. Opposite sides of a
rectangle are congruent to each
other.
Uniformity Theorem:
Corr 3: In a Hilbert plane satisfying
the acute (respectively, obtuse) angle
hypothesis, a side of a Lambert
quadrilateral adjacent to the acute
(respectively, obtuse) angle is greater
than (respectively, less than) its
opposite side.
Uniformity Theorem:
Corr 4: In a Hilbert plane satisfying the
acute(respectively, obtuse) angle hypothesis, the
summit of a Saccheri quadrilateral is greater than
(respectively, less than) the base. The midline
segment MN is the only common perpendicular
segment between the summit line and the base
line. If P is any point  M on the summit line and
Q is the foot of the perpendicular from P to the
base line, then PQ > MN (respectively, PQ < MN).
As P moves away from M along a ray of the
summit line emanating from M, PQ increases
(respectively, decreases).
Saccheri Angle Theorem:
For any Hilbert plane:
a) If there exists a triangle whose
angle sum is < 180, then every
triangle has an angle sum of < 180,
and this is equivalent to the fourth
angles of Lambert quadrilaterals and
the summit angles of Saccheri
quadrilaterals being acute.
(continued)
Saccheri Angle Theorem:
For any Hilbert plane:
b) If there exists a triangle whose
angle sum is = 180, then every
triangle has an angle sum of = 180,
and this is equivalent to the plane
being semi-Euclidean.
(continued)
Saccheri Angle Theorem:
For any Hilbert plane:
c) If there exists a triangle whose
angle sum is > 180, then every
triangle as an angle sum of > 180,
and this is equivalent to the fourth
angles of Lambert quadrilaterals and
the summit angles of Saccheri
quadrilaterals being obtuse.
Saccheri Angle Theorem:
Lemma:
Let ABDC be a Saccheri
quadrilateral with summit angle class
. Consider the alternate interior
angles ACB and DBC with respect to
diagonal CB:
a) ACB < DBC   is acute.
b) ACB  DBC   is right.
c) ACB > DBC   is obtuse.
.
Non-Obtuse-Angle Theorem:
A Hilbert plane satisfying
Aristotle’s axiom either is semiEuclidean or satisfies the acute
angle hypothesis (so that by
Saccheri’s angle theorem, the angle
sum of every triangle is  180).
(Corollary follows)
Non-Obtuse-Angle Theorem
Corr:
In a Hilbert plane satisfying
Aristotle’s axiom, an exterior angle of
a triangle is greater than or
congruent to the sum of the two
remote interior angles.
Saccheri-Legendre Theorem:
In an Archimedean Hilbert plane, the
angle sum of every triangle is  180.