1. Non –Euclidean Geometry
Chapter 4

2. 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.

3. 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.

4. 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'.

5. 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)

6. Thm 4.3: Measurement Thm (con’d)
3) If 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)

7. 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) is a positive real number and = 1.
8) = iff AB  CD.
9) A*B*C iff
10) < iff AB < CD.
11) For every positive real number x, there exists
a segment AB such that = x.

8. Corr 2: to EA Theorem:
The sum of the degree measures of any two angle of a triangle is less
than 180.

9. Triangle Inequality:
If A, B, and C are three noncollinear points, then
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.

10. 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 .

11. 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.)

12. 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

13. 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.)

14. Last Stop before Non-Euclidean Geometry!
Saccheri
And
Lambert
Quadrilaterals

15. 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) bi-right quadrilaterial.
See figure: 

16. Lambert Quadrilateral
A quadrilateral with at least three right angles is called a Lambert Quadrilateral.

17. 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.

18. 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

19. 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.)

20. 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.

21. Lambert Quadrilateral
A quadrilateral with at least three right angles is called a Lambert Quadrilateral.

22. 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.

23. 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.

24. 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.

25. Uniformity Theorem:
Corr 2: There exists a rectangle in a Hilbert plane  the plane is semi- Euclidean. Opposite sides of a rectangle are congruent to each other.

26. 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.

27. 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).

28. 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)

29. Saccheri Angle Theorem: For any Hilbert plane:
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)

30. 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.

31. 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. .

32. Non-Obtuse-Angle Theorem:
A Hilbert plane satisfying Aristotle’s axiom either is semi-Euclidean or satisfies the acute angle hypothesis (so that by Saccheri’s angle theorem, the angle sum of every triangle is  180). (Corollary follows)

33. 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.

34. Saccheri-Legendre Theorem:
In an Archimedean Hilbert plane, the angle sum of every triangle is  180.