Non-Euclidean
Geometry
Dr. Connie S. Schrock
MA 722
Sections 1.5 – 1.6
Consequences and Substitutions for the Fifth

Euclid’s Fifth Postulate
Postulate 5
That, if a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
Playfair’s Axiom
Through a point not on a given line there passes not more than
one parallel to the line.

High School Geometry Texts create their own set of Postulates to create the same
Geometry

Properties Implied by the Fifth
Postulate
The sum of the angles of a triangle equal two right angles.
Given
ABC with angles labeled as shown.
Construct the line h through A so that h ││
creating
angles 4 and 5.
By Prop. 29 m<5 = m<3 and m<2 = <4.
Using Prop. 13 we know that <1 + <4
+ <5 = two right
angles.
Using Axiom 1 (substitution) <1 + <2
+ <3 = two right
angles.
This is dependent on the fifth as we used a proposition after 28 that was dependent on the fifth.