# Points, Lines, and Triangles in Hyperbolic Geometry.

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Oct 10, 2013 (4 years and 9 months ago)

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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 unit
s 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 fol
lowing 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 intersec
tion 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 giv
en line.

2

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 supp
lementary.

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 180
o
.

8.

Exterior Angle Theorem:

The measure of an exterior angle of a triangle is equal to
t
he 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 triangl
es 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 tri
angles 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.

4

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
understan
ding 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 poi
nt 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.

Establis
h that the sum of the angles of a triangle on the hyperbolic plane is less than
180
0
.

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.

Inve
stigate 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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M
odels 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 a
s 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 p
lane. 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 t
hat 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
poin
ts 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 hy
perbolic plane.) The
Klein model satisfies the Hyperbolic Axiom of Parallelism.

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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 sat
isfies 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 compl
ex exercise.

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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 po
ints 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
m
odel 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 c
ircle 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.

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 a
ngle 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 dynamical
ly. 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 Sketchpad
1

of the users Geometer’s Sketchpad program. This
software uses the Poincaré model for
hyperbolic geometry and allows students to experiment drawing lines, triangles, bisecting
angles and lines, and much more. It is important that students understand that they are
studying aspects of hyperbolic geometry i
ntuitively through the use of models. While this
study is not exact, it does allow students the opportunity to gain an understanding of the
geometry and how it compares with Euclidean geometry on the plane. There are some
advantages of this software over t
he second one to be discussed later. Geometer’s
Sketchpad presents the Poincaré disk with its center clearly shown and the user is able to
move a line and observe what happens to points on the line as it is moved. In addition, if

between Euclidean and hyperbolic geometry, the student can
work in the Euclidean plane alongside the Poincaré disk, and comparisons can be easily
made. A disadvantage of this model is that it does not offer a script for reflection and
therefore this aspec
t of the geometry is difficult to study.

1

MarleleC Dwyer and Richard E. Pfiefer.
Exploring Hyperbolic Geometry with the

Mathematics Teacher Volume 92 Number 7 October 1999

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A second website offering students the opportunity to expe
rience hyperbolic
geometry dynamically can be found on the Internet at

http://math.rice.edu//~joel/NonEuclid

NonEuclid is a Java Sotware simulation that offers ruler and compass constructions using
b
oth the Poincaré Disk and the Upper Half
-
Plane models of hyperbolic geometry. The
program can be downloaded directly by clicking in the space indicated. If students wish
to use the Poincaré disk, they are presented with a circle representing the hyperbolic

space. The user can then select to plots points, find midpoints, find points of intersection,
or plot a point on an object. In addition, the following constructions are available: draw a
segment, ray or line, draw a perpendicular, draw a circle, bisect an

angle, and reflect. The
user can also select to draw a segment of specific length (useful for drawing isosceles or
equilateral triangles), or a ray at a specific angle.

One disadvantage of this software over the Geometer’s Sketchpad is that the disk
does

not have its center shown. An advantage is that it has a reflection option that offers
the user the opportunity to tessellate the plane. Reflection is also very useful when
investigating aspects of congruence of triangles on the hyperbolic plane. The stud
ent will
also find this construction site very useful when attempting to construct one angle
congruent to another or one line segment congruent to another. The option to
draw a
segment of a specific length
or to
draw a ray at a specific angle

is very usefu
l when
attempting to discover aspects related to congruence and similarity of triangles

Introduction.

What follows is a series of student
-
centered activities in which students are actively
involved in discovering similarities and differences between Eucl
idean geometry and
hyperbolic geometry.

Although the activities presented would work equally well on
either the disk or half
-
plane model, for the purposes of consistency, we will use only the
Poincaré Disk model.

The approximate time for each activity is

shown in parentheses next to the
Note to
the Teacher

following each activity. Students should be encouraged to do each activity

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as it arises and answer the accompanying questions. When students are asked to do a
construction, they should be encouraged to

actually do so as this serves as an opportunity
for students to review these construction methods. Likewise, when students are asked to
prove one of their Euclidean theorems, they should be encouraged to actually do this, as
they will once again be review
ing some of the properties of the Euclidean geometry they
have already learnt. The teacher may wish to take time out at the end of each period to
discuss the students’ observations and get feedback from the students on what they have
discovered.

Activitie
s

After students have been introduced to the Poincaré disk and lines in this model of
the hyperbolic plane, they are ready to use the software introduced above for the
following activities.

1.

In Euclidean geometry the
point

is an undefined term and is used

as a
foundation on which the geometry is developed. Do you think that we could
adopt the point as an undefined term in hyperbolic geometry? Justify your

Note to the teacher:
(5 minutes)

As in Euclidean geometry, we need some basic terminology on

which to build
hyperbolic geometry. The point can be adopted as an undefined term in hyperbolic
geometry. Students should construct some points on the Poincaré disk and label them
A, B, C, and so on.

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2.

a) Locate two points in the Euclidean plane. What

is the shortest path between
these two points?

b) Use the Euclidean point tool to locate two points A and B in the hyperbolic
plane. Using the hyperbolic segment tool, draw the segment between the two
points. Compare the segments drawn. In the Geometer’s
model, use the Euclidean select tool to move one of the endpoints of the segment
around the plane. In the Non
-
Euclid utility, consider a fixed point, and then
construct different lines which pass through this point and another point on t
he
disk. Comment on the nature of the line

segment

joining the points in the
hyperbolic plane as compared to the Euclidean line segment.

A line in the Poincaré disk becomes straight in the Euclidean sense when it passes
through the center of t
he disk.

Note to the teacher:
(10 minutes)

In hyperbolic geometry a
line

is defined as an arc of a circle that is orthogonal to
the circumference of the disk. It should be pointed out to students that while the
lines

in the hyperbolic plane appear to be

finite in length, this is in fact not the
case. Distances are distorted in this model, and the boundary of the disk is
considered to be at infinity. Students also need to note that when a line is
constructed it looks more curved when it is away from the c
enter of the disk and
as it becomes closer and closer to the center, it becomes more straight in the
Euclidean sense.

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Before going any further, it may be important that students realize that in the
Geometer’s Sketchpad model there are two different line t
ools and line segment
tools that are being used throughout these activities. When working on the
Euclidean plane, the Euclidean line tool is used. If this tool is used on the
hyperbolic plane, a Euclidean line segment will result. In order to draw lines or

line segments on the hyperbolic plane, the student must select the hyperbolic line
or line segment option.

3.

Euclid’s first postulate states that for every point P and for every point Q where
P

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䑲aw⁴漠p潩o瑳⁐⁡d⁑⁯n⁴桥⁤獫⸠䑲慷⁡
line

that passes
through these two points. Label two points on the line. Try to see if you can draw
a different line through these two points. Does Euclid’s first postulate hold in
hyperbolic geometry?

Note to the teacher.
(10 minutes)

Euclid’s first postulate holds in hyperbolic geometry. A unique line exists through
any two points in the hyperbolic plane.

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4.

a) Locate three points A, B, and C on a line on the Euclidean plane. The
Betweenness Axiom

st
ates that if A, B, and C are points on the Euclidean plane,
then one and only one point is between the other two.

b) Does the
Betweenness Axiom

hold on the hyperbolic plane?

Note to the teacher:
(5 minutes)

The Betweenness Axiom holds for hyperbol
ic plane.

5.

a) Draw two lines on the Euclidean plane. In how many points do these lines
intersect?

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b) Draw two lines on the hyperbolic plane. In how many points do the lines
intersect?

Note to th
e teacher.
(10 minutes)

On both the Euclidean and hyperbolic planes, lines intersect in a maximum of one
point, or they have no points of intersection (parallel lines).

6.

Two lines are defined as being parallel if they have no points in common.

a)

Draw a line

l

on the Euclidean plane. Locate a point A that is not on
l
.
Construct a line through A that is parallel to
l
. How many possible lines can
you construct?

b)

Draw a line
m

on the hyperbolic plane. Mark a point P not on
m
. Draw a line
through P that is
parallel to
m
. How many possible lines can you construct?
Does the Parallel Postulate hold on the hyperbolic plane?

15

Note to the teacher.
(15 minutes)

The student should discover that an infinite number of lines can be drawn through
P parallel t
o
m
. Euclid’s fifth postulate does not hold in hyperbolic geometry, and
is in fact what separates the two geometries.

7.

State Euclid’s
parallel postulate
. How would you re
-
word the postulate so that it
is true for the hyperbolic plane?

Note to the teacher:

(10 minutes)

The equivalent of the parallel postulate on the hyperbolic plane states that if
l

is a
line and P is a point not on
l

then more than one line parallel to
l

can be drawn
through P.

8.

Lines in Euclidean geometry are of infinite length. Can the s
ame be said of lines
in the hyperbolic plane?

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Note to the teacher.
(10 minutes)

Students should be reminded that although the Poincare

model uses a disk to
represent the hyperbolic plane, the boundary of the disk represents infinity in

the
hyperbolic sense. This means that if a two
-
dimensional creature existed in the
center of the disk and this creature walked towards the boundary of the disk with
steps of equal length, then for an observer on the outside, it would seem that the
steps w
ere getting progressively shorter and shorter. This means that distances are
distorted in the Euclidean sense in this model. In the above diagram the distance
from P to Q is 1.94 units while the distance from S to T is 3.38 units, yet the
distance from S t
o T appears to be much less than that from P to Q

9.

a) Euclid’s third postulate states that a circle can be drawn with any center and

b) Draw a number of circles with centers located at different points in the
hyperbolic plane. What appears to
happen to the circle as the center gets nearer
the edge of the disk? Does this mean that the center of a circle near the edge of
the disk is not located equidistant from the points on its circumference?

17

Note to the teacher.
(10

minutes)

Students should be reminded that Euclidean distances are not conserved in the
hyperbolic plane. All points on the circumference of the circle in the hyperbolic
plane are the same constant distance from the center of the circle. This is evident
fr
om the second diagram shown above.

10.

a) Draw two intersecting lines on the Euclidean plane. Use a protractor to
measure the vertical angles. Confirm that the vertical angles are congruent.

b) Draw two lines on the hyperbolic plane.

i)

Measure th
e pairs of adjacent angles. Are they supplementary?

ii)

Measure the vertical angles. Are the pairs of vertical angles
congruent?

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Note to the teacher:
(15 minutes)

Students will discover that the adjacent angles on a hyperbolic line are
supplement
ary and that the vertical angles are congruent.

11.

a) Draw a line on the Euclidean plane. Locate a point A not on the line.
Construct a perpendicular from point A to the line. How many perpendiculars
can you construct?

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b) Draw a line on the hype
rbolic plane. Locate a point P not on the line. Can you
construct a perpendicular from the point to the line? If so, how many
perpendiculars can you construct?

Measure the angle at the point of

intersection to confirm that the

angle is a right angle.

Note to the teacher:
(15 minutes)

On both the Euclidean plane and the hyperbolic plane only one perpendicular can
be drawn from a point to a line. The construction of a perpendicular in the
hyperbolic plane is done in much the same way as one would const
ruct a
perpendicular on the Euclidean plane. Locate a point P not on line
l
. Describe a
circle with P as center to cut
l

in points M and N. Describe a circle with M as
center and passing through P. Draw a circle with N as center and passing through
P. Mark

the other point at which the circle intersect with Q. Join P and Q. PQ is
perpendicular to
l
.

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12.

a) Draw a pair of parallel lines and a transversal on the Euclidean plane.
Measure the corresponding angles and confirm that they are congruent.

b) Draw a pair of parallel lines and a transversal on the hyperbolic plane.
Measure the pairs of corresponding angles and determine whether the
corresponding angles postulate

is valid on the hyperbolic plane.

Note to the teacher:
(10 minutes
)

Students will discover that on the hyperbolic plane, corresponding angles are not

congruent. Here it is important that students are reminded that lines are parallel if
they do not have common points. In the hyperbolic plane, parallel lines are not
equid
istant from each other.

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13.

a) Draw a pair of parallel lines on the Euclidean plane. Prove that the alternate
interior angles are congruent.

b) Draw a pair of parallel lines on the hyperbolic plane. Measure the alternate
angles and determine whether th
ey are congruent.

Note to the teacher:
(10 minutes)

Student will discover that on the hyperbolic plane, alternate angles are not
congruent.

14.

a) Draw a pair of parallel lines on the Euclidean plane. Prove that the
consecutive interior angles are
supplementary.

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b) Draw a pair of parallel lines on the hyperbolic plane. Measure the consecutive
interior angles. Are these pairs of angles supplementary?

Note to the teacher:
(10 minutes)

The student will discover that the consecutive i
nterior angles on the hyperbolic
plane are not supplementary.

15.

The Perpendicular Transversal Theorem states that if a transversal is
perpendicular to one of two parallel lines on the Euclidean plane, then it is
perpendicular to the other.

Draw two paralle
l lines
l

and
m

on the hyperbolic plane. At a point on
l

draw a
perpendicular transversal. Determine whether the above theorem is valid on the
hyperbolic plane.

23

Note to the teacher:
(10 minutes)

The student will discover that in some cas
es a transversal for one of a pair of
parallel lines does not intersect the second line. When the transversal does
intersect, it is not perpendicular to the second line. The Euclidean theorem is thus
not valid on the hyperbolic plane.

16.

a) Draw line
l
on t
he Euclidean plane. Through point A, not on
l, construct

a line
m

that is parallel to
l
. Locate another point B that is not on either
l
or
m
. Draw a
line
n

through B parallel to
l
. Is
m

parallel to
n
? Can you prove this?

24

b) Draw a line
r

on the hyp
erbolic plane. Through a point P not on
r
, draw a line
s

that is parallel to
r
. Through point Q that is not on either
r
or
s
, draw a line
t

that is parallel to
r
. Are
s

and
t

parallel?

Figure (a)

Figure (b)

Note to the teacher:
(10 minutes)

Students will confirm that on the Euclidean plane if two lines are parallel to the
same line, then they are parallel to each other. On the hyperbolic plane, however,
it is possible for two lines parallel to a third line to be parallel or non
-
p
arallel as
shown in the diagram above.

In Figure (a) above, r || s and r || t, and s || t. However, in figure (b) r || s and r || t
and s and t are not parallel.

17.

a) Draw a line
l

on the Euclidean plane. Locate two points A and B on the line.
At each point

construct a perpendicular line. Are the two perpendicular lines
parallel. Can you prove this?

25

b) Draw a line
m

on the hyperbolic plane. Locate at least two points P and Q on
the line. At each point draw a perpendicular to the line. Are the two lines
para
llel?

c) On the Euclidean plane if two lines are perpendicular to the same line, then
the two lines are perpendicular. Is this true for the hyperbolic plane?

Note to the teacher:
(15 minutes)

Students will discover that, as for the Euclidean p
lane, if two lines are
perpendicular to the same line on the hyperbolic plane, then the two perpendicular
lines are parallel.

In neutral geometry (the geometry without any parallelism axiom) which is true
for both Euclidean and hyperbolic geometry, we are
able to prove numerous
theorems. One of these theorems is the
Alternate Interior Angle Theorem.

This
theorem states the following:

If two lines cut by a transversal have a pair of congruent alternate interior
angles, then the two lines are parallel.

The p
roof of this theorem proceeds as follows. This proof may be difficult for
some students to understand, but it is provided none the less for those students
who may enjoy seeing the proof.

26

Consider two lines

l

and
l’

and a transversal as shown
in the figure, and let

and

be the two alternate interior angles that are congruent. Now, if
l

and
l’

are not parallel
then they should meet at a point such as B in the figure. We now find point C on
l

on the
opposite of B such that
. Then we see that
. In
particular,
. Therefore, since

and

are supplementary,

and

should be supplementary. This means that C lies on
l
, and hence
l

and
l’

have two points
in common, which contradicts the first axiom in both geometries, namely that two
distinct points define a unique li
ne. Therefore,
l

||
l’
.

A corollary to this theorem is that two lines that are perpendicular to the same line are
parallel. This is true because if
l

and
l’

are both perpendicular to t, the alternate interior
angles are right angles and they are congruent.

Therefore we have proved in both geometries that if two lines that are perpendicular to a
line then they are parallel.

27

18.

When two lines cut by a transversal on the Euclidean plane, have congruent
corresponding angles, then the two lines are parallel.

Investigate whether the same is true for lines drawn on the hyperbolic plane.

Note to the teacher:
(15 minutes)

Students will discover that when corresponding angles are congruent, lines on the
hyperbolic plane will be parallel.

In fact, this

is another corollary to the Alternate Interior Angle Theorem in neutral
geometry that we have just completed the proof for. In the following figure

and
are corresponding angles.

and

are vertical angles that are
congruent in both geometries. Therefore, if
, then
. These
two are alternate interior angles and by the Alternate Interior Angle Theore
m we
conclude that
l

and
l'
’are parallel.

28

19.

a) Draw triangle ABC on the Euclidean plane. Prove that the sum of the interior
angles of a triangle on the Euclidean plane is equal to 180
o
.

b) Draw a triangle on the hyperbolic plane. Use th
e hyperbolic measure tool to
measure the interior angles of the triangle. Compare the sum of the angles of a
hyperbolic triangle to that of a Euclidean triangle.

29

Note to the teacher:
(15 minutes)

Based on their background in Euclidean geometry,

students should be able to present
a sketch and proof to show that the sum of the angles of a triangle on the plane is
180
o
. Note that in the diagram provided above,
∆ABC is arbitrary, and line
m

is
parallel to segment AB.

and

are alternate interior angles, and therefore
congruent,

and

are corresponding angle
s and are therefore congruent, and
the sum of the three angles
,

and

is 180
0

.

For the hyperbolic case, students realize that

(a)

the sum of the angles of a hyperbolic triangle

is less than 180
0

(b)

there is not any fixed value (such as 180
0
) for the sum of the angles of a
hyperbolic triangle.

Students will also discover that they are able to construct triangles with angle
measure of zero degrees. Since these triangles have their ve
rtices on infinity, we
consider them as a special case of study that is beyond the scope of this current work.

20.

a) Draw a triangle on the Euclidean plane. Extend one side of the triangle to
create an exterior angle. Prove that the exterior angle is equal t
o the sum of the
two non
-

30

b) Draw a triangle on the hyperbolic plane. Extend one of the sides of the
triangle. Measure the exterior angle and compare this measure with the measure
of the sum of the measure of the two no
n
-
adjacent interior angles. What can you
conclude?

Note to the teacher:
(15 minutes)

Students can prove that the exterior angle of a triangle is equal to the sum of the
two remote interior angles by using the triangle as shown above. Since the
sum of
the angles of a triangle is 180
o
,
. But the adjacent angles at
C are supplementary, so
. Therefore,

The student will discover that the measure of the exterior angle of

a hyperbolic
triangle is not equal to the sum of the measures of the non
-
angles. Logically, the student may argue that since

and

is less than 180
o
, then

is more than the sum of

and
. Moving from intuition to logic is important and teachers should encourage
students to think about possible reasons for their answers before using the
software to confirm t

31

21.

On the Euclidean plane, if two angles of one triangle are congruent to two angles
of another triangle, then the third angles are congruent.

Draw a triangle on the hyperbolic plane. Measure the angles of the triangle.
Create a second trian
gle with two angles in the second triangle congruent to two
angles in the first. Measure the third angle of the triangle. Are the third angles
congruent?

Note to the teacher.
(15 minutes)

Students may find it easier to use the NonEuclid websi
te for this activity. This
website offers a construction to create one angle congruent to another, making it
easier for the student to create the required triangles.

Students will discover that if two angles of one triangle on the hyperbolic plane
are cong
ruent to two angles of another, then the third angles are not necessarily
congruent. Here, it is important to keep in mind that the third angles are not
congruent unless two triangles are congruent. This means that AAA is sufficient
to prove that two trian
gles in the hyperbolic plane are congruent. This point will

32

22.

a) Draw an isosceles triangle on the Euclidean plane. Prove that the angles at the
base of the congruent sides are congruent.

b) Draw an
isosceles triangle on the hyperbolic plane. Measure the angles at the
base of the congruent sides. Are the base angles congruent?

Note to the teacher:
(15 minutes)

The proof for the Euclidean plane can be achieved by proving that
∆ABC is
congruent to ∆ACB by SSS where the corresponding congruent sides are
. Corresponding angles are therefore congruent,
and
.

33

In the hyperbolic case, the NonEuclid program once again offers student
s an easy
way to construct lines of equal length. Using this program will enable them to
discover that the base angles theorem is valid on the hyperbolic plane. We may
mention that SSS is also true in hyperbolic geometry and therefore a similar proof
is va
lid in hyperbolic geometry.

23.

a) Draw a triangle on the Euclidean plane with two angles congruent. Prove that
the sides opposite the congruent angles are congruent.

b) Draw a triangle with two angles congruent on the hyperbolic plane. Measure
the

sides opposite the congruent and report whether these sides are congruent.

Note to the teacher.
(15 minutes)

On the plane the students should be able to prove that
using ASA,
and therefore
.

As in Euc
lidean geometry, if two angles of a triangle on the hyperbolic plane are
congruent, then the sides opposite the angles are congruent.

34

24.

a) Draw an equilateral triangle on the Euclidean plane. Prove that the measure
of each angle of an equi
lateral triangle is 60
0

b) Draw an equilateral triangle on the hyperbolic plane. Determine the measure
of each angle of the equilateral triangle. How do your observations on the
hyperbolic plane compare with those on the Euclidean plane?

35

Note

to the teacher:
(15 minutes)

Students should be able to construct an equilateral triangle using a compass and a
straight edge only, given a side of the triangle. Let

be the side of the triangle.
If we now construct two circles
with the same radius as AB, one with center at A
and the other centered at B, these two circles will intersect in two points C and C`.
Each of these two vertices with A and B will create equilateral triangles.

Students should use the same met
hod to create equilateral triangles on the
Poincaré disk. They will notice that their equilateral triangles don’t seem to have
congruent sides. Once more, students should be reminded that distances seem to
be distorted on the Poincaré disk in our Euclidean

eyes.

36

Students will discover that equilateral triangles on the hyperbolic plane are also
equiangular. Unlike Euclidean equilateral triangles, however, each angle is not
equal to 60
o
, and is different from one triangle to another.

25.

a)
What is the formula for calculating the area of a triangle on the Euclidean
plane?

b) Investigate whether this formula is valid for calculating the area of a triangle
on the hyperbolic plane.

37

Note to the teacher:
(15 minutes)

The formula for the
area of a triangle on the Euclidean plane is A = ½bh where
b
is the base of the triangle and
h
is the height. When we study a triangle on the
Poincaré disk, a different value is obtained for each calculation of
1
/
2

bh for each
pair of base and height used
. This formula can therefore not be used to calculate
the area of a triangle on the hyperbolic plane.

26.

In a right triangle on the Euclidean plane, the square on the hypotenuse is equal
to the sum of the squares on the legs of the triangle. D
oes this Theorem of
Pythagorean hold for triangles on the hyperbolic plane?

Construct a number of right triangles on the hyperbolic plane. Use the
hyperbolic measure segment option to measure the lengths of the hypotenuse
and legs. Use the calculate comman
d under the measure command to discover
whether this theorem is valid on the hyperbolic plane.

38

Note to the teacher:
(20 minutes)

The following figure presents a right triangle on the hyperbolic plane in the
Poincaré disk. Students will discover that the

Theorem of Pythagoras is not valid
on the hyperbolic plane.

Note to the teacher:

In general, when we say that two triangles

and

are congruent,
we mean that if the two triangles are dr
awn in two different locations, then we can
move one triangle so that it will coincide exactly with the other. For students, it is
sufficient that they understand that two triangles are congruent if all their
corresponding angles and corresponding sides ar
e congruent. In Euclidean
geometry we realize that having SSS guarantees that all angles are congruent as
well, and therefore the two triangles are congruent. The same is true for SAS and
ASA. For the following questions, what we would like to do is to cre
ate two
triangles in the hyperbolic plane based on any of the SSS, SAS, or ASA
conditions, and then check to see if all other congruence relations exist (all
corresponding angles are congruent and all corresponding sides are congruent).

[

39

27.

The SAS Congrue
nce Postulate states that 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.

Investigate whether this postulate can be accepted on th
e hyperbolic plane.

Note to the teacher:
(20 minutes)

Students will construct

on the disk. Then, using the tool measure, they will
construct

congruent to

in an
other location. They will then use the angle
measure tool to construct an angle on DE with vertex D which is congruent to
angle A. Now, on this new side of the angle, find a point F such that
.
We notice that these two triangles
have two sides and an angle between them
congruent
. Now what we need to do is to check
that all other corresponding components are congruent as well

(i.e.
). Students will find through measurement
that

all other corresponding components are congruent.

40

28.

The SSS Congruence Theorem and the ASA Congruence Theorems are valid on
the Euclidean plane. Use the NonEuclid webiste to discover whether these
theorems are valid on the hyperbolic plane

No
te to the teacher:
(20 minutes)

Once again, students will construct

on the disk. Using the tool measure,
construct
. What we need to do now is determine
whether the corresponding angles are congruent.
Students should use the tool to
measure angles to confirm that the corresponding angles are congruent.

29.

On the Euclidean plane, if three angles of one triangle are congruent to three
angles of another triangle, then the corresponding sides of the triangles

are in
proportion and the two triangles are similar.

If three angles of one triangle on the hyperbolic plane are congruent to three
angles of another, what can we say about these two triangles?

41

Note to the teacher:
(20 minutes)

Construc
t

on the hyperbolic plane. Use the angle measure tool to measure
the size of each angle in the triangle. Draw a segment DE and at vertex D
construct an angle that is congruent to
. At vertex E construct

an angle that is
congruent to the angle at B. The student will discover the third angle of

is
only congruent to the third angle of

when the two triangles are identical,
i.e. the two triangles are con
gruent. Similar triangles do not exist on the
hyperbolic plane unless the two triangles are congruent; in which case they are
identical.