Study Guide
Timeline
• Euclid’s five axioms (300 BC)
• From Proclus (400AD) belief that the fifth axiom is
derivable from the first four
• Saccheri (17th century): Either HOA or HAA or HRA. Also
the first four axioms imply either HRA or HAA.
• Discovery of non-Euclidean Geometry
(1st half of the 19th century) by Gauß, Bolyai and
Lobachevsky (1829)
– The fifth axiom is independent and cannot be derived from the
first four.
– There is a dichotomy: either one or two Lobachevsky parallels
• Models for non-Euclidean geometry (2nd half of the 19th
century) by Beltrami (1868), Klein (1871) and Poincaré.
Study Guide
Triangles
• HOA <=> Sum of interior angles of a
triangle is > 180°
• HAA <=> Sum of interior angles of a
triangle is < 180°
• HRA <=> Sum of interior angles of a
triangle is = 180°
Study Guide
Equivalent versions of the Parallel Postulate
•
•
•
That, if a straight line falling on two straight
lines make the interior angle 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 two right
angles. (Euclid ca. 300BC)
For every line l and for every point P that does
not lie on l there exists a unique line m through
P that is parallel to l. (Playfair 1748-1819)
The sum of the interior angles in a triangle is
equal to two right angles. (Legendre 17521833 )
Study Guide
Angles and the Parallel Postulate
Assuming the first four axioms there is a dichotomy:
Euclidean or non-Euclidean
1. Euclidean: equivalently
1.
2.
2.
The sum of the interior angles of a triangle is 180.
There is exactly one parallel to a given line through a point not
on that line.
Non-Euclidean: equivalently
1.
2.
3.
The sum of the interior angles of a triangle is strictly less than
180.
There is exactly two parallel lines, in the sense of
Lobachevsky, to a given line through a point not on that line.
To a given line there are infinitely many lines through a through
a point not on that line which do not intersect this line.
Study Guide
Facts
The dichotomy means that
If one triangle has the sum of interior angles equal to 180° the so
do all triangles
And
If one triangle has the sum of interior angles strictly less than 180°
the so do all triangles
Also
If there is one line which through a given point has only one parallel
than this statement is true for all lines and points.
And
If there is one line which through a given point has more than one
parallel than this statement is true for all lines and points.
Sample Questions
• By whom and when was non-Euclidean
geometry discovered?
• State Euclid’s version of the parallel postulate?
• Give other equivalent formulations!
• Which postulate equivalent to the parallel
postulate did Legendre assume in his false
deduction?
• State the first four axioms of Euclid!
• Show that the parallel postulate implies that the
sum of the interior angels in a triangle is 180°!
Sample Questions
• Show the equivalence of Euclid’s parallel postulate and
the Playfair’s version which states that there is a unique
parallel.
• Draw a picture of a parallel at angle 45° in a model of
non-Euclidean geometry!
• What values can the sum of interior angles of a triangle
take in a non-Euclidean geometry?
• What is the contribution of Beltrami to non-Euclidean
geometry?
• What is Gauss’s contribution?
• Is there a relationship between area and angles of a
triangle in Euclidean geometry? How about in nonEuclidean geometry.
And study the homework!