Points, Lines, and Triangles in Hyperbolic Geometry. Postulates and Theorems to be Examined. In forming the foundation on which to build plane geometry, certain terms are accepted as being undefined, their meanings being intuitively understood. The units that are presented will accept the following undefined terms: Point Line Lie on Between Congruent. Terms used in the modules will be defined as follows: 1. Line segment: The segment AB, , consists of the points A and B and all the points on line AB that are between A and B 2. Circle: The set of all points, P, that are a fixed distance from a fixed point, O, called the center of the circle. 1
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Points, Lines, and Triangles in Hyperbolic Geometry.
Postulates and Theorems to be Examined.
In forming the foundation on which to build plane geometry, certain terms are
accepted as being undefined, their meanings being intuitively understood. The units that
are presented will accept the following undefined terms:
Point
Line
Lie on
Between
Congruent.
Terms used in the modules will be defined as follows:
1. Line segment: The segment AB, , consists of the points A and B and
all the points on line AB that are between A and B
2. Circle: The set of all points, P, that are a fixed distance from
a fixed point, O, called the center of the circle.
3. Parallel lines: Two lines, l and m are parallel if they do not intersect.
The following postulates will be examined:
1. There exists a unique line through any two points.
2. If A, B, and C are three distinct points lying on the same line, then one and only one
of the points is between the other two.
3. If two lines intersect then their intersection is exactly one point.
4. A line can be extended infinitely.
5. A circle can be drawn with any center and any radius.
6. The Parallel Postulate: If there is a line and a point not on the line, then there is
exactly one line through the point parallel to the given line.
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7. The Perpendicular Postulate: If there is a line and a point not on the line, then there
is exactly one line through the point perpendicular to the given line.
8. Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then
the pairs of corresponding angles are congruent.
9. Corresponding Angles Converse: If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines are parallel.
10. SAS Congruence Postulate: If two sides and the included angle of one triangle are
congruent respectively to two sides and the included angle of another triangle, then
the two triangles are congruent.
The following theorems will be explored:
1. Vertical Angles Theorem: Vertical angles are congruent.
2. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are congruent.
3. Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are supplementary.
4. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other.
5. Theorem: If two lines are parallel to the same line, then they are parallel to each
other.
6. Theorem: If two lines are perpendicular to the same line, then they are parallel to
each other.
7. Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle
is 180o.
8. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two nonadjacent interior angles.
9. Third Angles Theorem: If two angles of one triangle are congruent to two angles of
another triangle, then the third angles must also be congruent.
10. Angle-Angle Similarity Theorem: If two triangles have their corresponding angles
congruent, then their corresponding sides are in proportion and they are similar.
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11. Side-Side-Side (SSS) Congruence Theorem: If three sides of one triangle are
congruent to three sides of a second triangle, then the two triangles are congruent.
12. Angle-Side-Angle (ASA) Congruence Theorem: If two angles and the included side
of one triangle are congruent to two angles and the included side of a second triangle,
then the two triangles are congruent.
13. Theorem of Pythagoras: In a right triangle, the square on the hypotenuse is equal to
the sum of the squares of the legs.
14. Base Angles Theorem: If two sides of a triangle are congruent, then the angles
opposite the sides are congruent.
15. Converse of the Base Angles Theorem: If two angles of a triangle are congruent,
then the sides opposite them are congruent.
16. Equilateral Triangle Theorem: If a triangle is equilateral, then it is also
equiangular.
Finally, students will investigate whether they can use the formula 1/2 bh to find the area
of a triangle on the hyperbolic plane.
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Points, Lines, and Triangles in Hyperbolic Geometry.
Objectives:
During this module of activities, students will
1. Learn to use different software programs that enable them to gain an intuitive
understanding of hyperbolic geometry through the use of the Poincaré model that
supports the properties of hyperbolic geometry.
2. Compare their understanding of the terms point, line, and parallel in Euclidean
geometry with what they discover in hyperbolic geometry.
3. Determine which of Euclid’s five postulates are valid in hyperbolic geometry.
4. Determine whether the postulate of betweenness holds in hyperbolic geometry.
5. Determine whether vertical angles are congruent on the hyperbolic plane.
6. Determine that through a point not on a line, more than one parallel line can be drawn
to a given line.
7. Discover whether the theorems and postulates regarding corresponding, alternate, and
interior angles on the same side of the transversal are valid on the hyperbolic plane.
8. Establish that the sum of the angles of a triangle on the hyperbolic plane is less than
1800.
9. Determine whether the measure of the exterior angle of a triangle on the hyperbolic
plane is equal to the sum of the measures of the two nonadjacent interior angles.
10. Investigate the Third Angles Theorem.
11. Investigate similarity of triangles on the hyperbolic plane.
12. Investigate congruence of triangles on the hyperbolic plane.
13. Determine whether the base angles theorem and its converse are valid on the
hyperbolic plane.
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Models for Studying Hyperbolic Geometry.
Models are useful for visualizing and exploring the properties of geometry. A
number of models exist for exploring the geometric properties of the hyperbolic plane. It
should be pointed out to the students however, that these models do not “look like” the
hyperbolic plane. The models merely serve as a means of exploring the properties of the
geometry.
The Beltrami-Klein Model for Studying Hyperbolic Geometry.
The Beltrami-Klein model is often referred to simply as the Klein model because
of the extensive work done in geometry with this model by the German mathematician
Felix Klein. In this model, a circle is fixed with center O and fixed radius. All points in
the interior of the circle are part of the hyperbolic plane. Points on the circumference of
the circle are not part of the plane itself. Lines are therefore open chords, with the
endpoints of the chords on the circumference of the circle but not part of the plane.
The Hyperbolic Axiom of Parallelism states that for every line l and every point P
with P l there exists at least two distinct lines parallel to l that pass through P. Students
should be reminded at this stage that lines are defined as being parallel if they have no
points of intersection. From the figure it is clear that neither line n nor m meet l, and they
are thus both parallel to l. (The fact that the lines may intersect l outside the circle is of no
concern, since points outside the circle do not form part of the hyperbolic plane.) The
Klein model satisfies the Hyperbolic Axiom of Parallelism.
l
n
m
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In addition, it can be easily shown that the model satisfies the axioms of
incidence, betweenness, and continuity, and with more effort, it can be shown that the
model satisfies the axioms of congruence.
The Poincaré Half Plane Model for Studying Hyperbolic Geometry.
In the Poincaré half plane model, the Euclidean plane is divided by a Euclidean
line into two half planes. It is customary to choose the x-axis as the line that divides the
plane. The hyperbolic plane is the plane on one side of this Euclidean line, normally the
upper half of the plane where y > 0. In this model, lines are either
a) the intersection of points lying on a line drawn vertical to the x-axis and the half
plane, or
b) points lying on the circumference of a semicircle drawn with its center on the x-
axis.
Lines in the Poincaré Half Plane model
The model satisfies all the axioms of incidence, betweenness, congruence,
incidence and the hyperbolic axiom of parallelism. Angles are measured in the normal
Euclidean way. The angle between two lines is equal to the Euclidean angle between the
tangents drawn to the lines at their points of intersection. Finding the length of a line
segment is a more complex exercise.
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l
m
nP
x
A
B
C
x
B
AC
Finding angle measure in the Poincaré Half Plane model
The Poincaré Disk Model for Studying Hyperbolic Geometry.
Henri Poincaré (1854 – 1912) developed a disk model that represents points in the
hyperbolic plane as points in the interior of a Euclidean circle. In this model, lines are not
straight as the student is used to seeing them on the Euclidean plane. Instead, lines are
represented by arcs of circles that are orthogonal to the circle defining the disk. In this
model therefore, the only lines that appear to be straight in the Euclidean sense are
diameters of the disk. In addition, the boundary of the circle does not really exist, and
distances become distorted in this model. All the points in the interior of the circle are
part of the hyperbolic plane. In this plane, two points lie on a “line” if the “line” forms an
arc of a circle orthogonal to C. The only hyperbolic lines that are straight in the Euclidean
sense are those that are diameters of the circle.
mB
A
C
Lines in the Poincaré model Constructing the angle between two
lines in the Poincaré model.
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This model satisfies all the axioms of incidence, betweenness, congruence, continuity,
and the hyperbolic axiom of parallelism. The angle between two lines is the measure of
the Euclidean angle between the tangents drawn to the lines at their points of intersection.
Hyperbolic Software.
There are two programs that will allow students to discover aspects of hyperbolic
geometry dynamically. The first of these is a program of script tools created by Mike
Alexander and modified by Bill Finzer and Nick Jackiw for the Geometer’s Sketchpad1
The software can be downloaded from the Internet at the following address: