Download 2.7 to 3.2 - Oregon State University

Read 2.7, 3.1, 3.2
Section 2.7, Non Euclidean
Geometries
Hyperbolic Parallel Postulate:
There exists a line  and a point P
such that at least two distinct lines
pass through P that are parallel to
.
Poincare Half Plane Model:
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Consequences:
 Through a given point not on 
there are infinitely many parallels
 There are no triangles with angle
sum 180 
 There exist no triangles that are
similar but not congruent
 There exist triangles that cannot be
circumscribed
 The sum of the interior angles of a
triangle varies and is always less
than 180 
 No rectangles exist
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 There is an upper limit to the area
of a triangle
 The larger a triangles area, the
smaller its angle sum
 The distance between certain pairs
of parallel lines approaches 0 in
one direction and  in the other
 If two parallel lines are crossed by
a transversal, alternate interior
angles may or may not be
congruent
Elliptic Parallel Postulate: Given
any line  and point P not on  there
is no line through P parallel to 
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3.2 Neutral Geometry
Notation: AB, AB or AB , ABC,
mABC, P  S  Q, AB
Def. Two segments are congruent
iff their measures are equal, two
angles are congruent iff their
measures are equal.
Def. Two polygons are congruent
iff there exists a 1-1
correspondence between vertices
such that all corresponding sides
and angles are congruent.
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Thm: The relations of line segment
congruence, angle congruence,
and polygon congruence are
equivalence relations.
Thm: Pasch’s Axiom If a line 
intersects PQR at a point S such
that P  S  Q
Proof:
Thm: Every line segment has
exactly one midpoint. Every angle
has exactly one bisector.
The Crossbar Thm: If X is a point
in the interior of UVW, then UX
intersects WV at a point Y such that
W  Y  V.
Thm: Supplements and
complement of the same or
congruent angles are congruent.
Isoceles Triangle Thm: If two
sides of a triangle are congruent,
then the angles opposite those
sides are also congruent.
Thm: Vertical angles are .
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Thm: A point is on the
perpendicular bisector of a line
segment iff it is equidistant from
the endpoints of that segment.
Def. An angle that is both
supplementary and adjacent to an
angle of a triangle is called an
exterior angle.
Exterior Angle Theorem: An
exterior angle of a triangle is
greater in measure than either of
the nonadjacent interior angles.
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