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.
B E. Then ABC DEF.
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.
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
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.
( A) + ( B) + ( C) = 180.
(The angle sum of every triangle is 180 if we assume Hilbert’s Euclidean parallel postulate.)
Corr: 1, 2 & 3 follow
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.
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.
c) ACB > DBC is obtuse.