Playfair’s Axiom equivalent to
Euclid’s Fifth Postulate
Proofs

A Closer Look At Postulate 5
•
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.
These lines
will meet

Proving a Statement Equivalent to the Fifth
Postulate
•
Euclid’s Elements: Book 1
–
Definitions, postulates, common notions, and
propositions 1 -28.(Basis E).
•
2 Lemmas and a Theorem:
–
Lemma 1
: Lines intersect at a unique point
–
Lemma 2
: If
l
||
m
, then all points on
l
are on the
same side of
m
.
–
Theorem 1
: Two lines perpendicular to the same
line are parallel

EV
Playfair’s Axiom
⇔
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.
⇔
Through a point not on a given line
there passes not more than one parallel to the
line.