module for hyperbolic geometry

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.

1

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.

2

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.

3

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.

4

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

5

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.

6

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.

7

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:

http://mathforum.org/sketchpad/gsp.gallery/poincare/poincare.html

Instructions for the download are given and once downloaded, the scripts become a part

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 intuitively 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 the 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

comparisons are to be made 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 aspect of the geometry is difficult to study.

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

Geometer's Sketchpad. Mathematics Teacher Volume 92 Number 7 October 1999

8

http://mathforum.org/sketchpad/gsp.gallery/poincare/poincare.html

A second website offering students the opportunity to experience 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

both 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 student 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 useful 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 Euclidean 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

9

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

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 reviewing 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.

Activities

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

answer.

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.

10

A

C D

B

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 Sketchpad Poincaré

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 the

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 the disk.


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 center of the disk and

as it becomes closer and closer to the center, it becomes more straight in the

Euclidean sense.

11

A

Before going any further, it may be important that students realize that in the

Geometer’s Sketchpad model there are two different line tools 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 Q, a unique line passes through P and Q.

Create a new disk. Draw two points P and Q on the disk. Draw a 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.

12

P

Q

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

Betweenness Axiom states 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?


The Betweenness Axiom holds for hyperbolic plane.

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

intersect?

13

A BC

P QR

P Q R

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

intersect?


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?

14

l

A


The student should discover that an infinite number of lines can be drawn through

P parallel to 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?


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 same be said of lines

in the hyperbolic plane?

15

m

P


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 were 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 to 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

any radius.

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?

16

PT

SRQ

PQ = 1.94

PR = 2.56

PS = 3.18

PT = 5.33

ST = 3.38


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

from 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 the pairs of adjacent angles. Are they supplementary?

ii) Measure the vertical angles. Are the pairs of vertical angles

congruent?

17

Distance = 1.91

Distance = 1.91

Distance = 1.91

A

BC

D

E


Students will discover that the adjacent angles on a hyperbolic line are

supplementary 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?

18

R

SP

Q

O1

2

34

m1 = 70.0°

m2 = 70.0°

m3 = 110.0°

m4 = 110.0°

A

b) Draw a line on the hyperbolic 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.


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 construct 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.

19

P

m1 = 90.0° 1

l

P

NM

Q

m1 = 90

1

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.


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

equidistant from each other.

20

P Q

R

S

T

U

V

W

107.8

84.4

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 they are congruent.


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.

21

P Q

R

S

T

U

V

W

107.8

84.4

b) Draw a pair of parallel lines on the hyperbolic plane. Measure the consecutive

interior angles. Are these pairs of angles supplementary?


The student will discover that the consecutive interior 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 parallel 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.

22

P Q

R

S

T

U

V

W

107.8

95.6


The student will discover that in some cases 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 the 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?

23

l

m

m1 = 90.0°

1

m2 = 35.9°

2

n

m

lA

B

b) Draw a line r on the hyperbolic 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)


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-parallel 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?

24

r

s

t

P

Q

r

s

t

P

Q

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

parallel?

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?


Students will discover that, as for the Euclidean plane, 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 proof 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.

25

m

PQ

R

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 line. 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.

26

l

l'

B

A'

A

C

1

12

2

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.


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 Theorem we

conclude that l and l'’are parallel.

27

m1m1 = 90.0°

2

m2 = 90.0°

m3 = 90.0°

3

l'

l

A'

A

1

1

2

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 180o.

b) Draw a triangle on the hyperbolic plane. Use the 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.

28

Q

P

Rm2 = 12.9°

m3 = 11.6°

m1 + m2 + m3 = 42.2°

1

2

3

m1 = 17.7°

l m

A C

B

12

3


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

180o. 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 angles and are therefore congruent, and

the sum of the three angles , and is 1800 .

For the hyperbolic case, students realize that

(a) the sum of the angles of a hyperbolic triangle is less than 1800

(b) there is not any fixed value (such as 1800) 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 vertices 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 to the sum of the

two non-adjacent interior angles.

29

A

B DC1 2

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 non-adjacent interior angles. What can you

conclude?


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 180o, . 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-adjacent interior

angles. Logically, the student may argue that since and

is less than 180o, 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 their answers.

30

Q

P

SR1 2

P=14.2°

Q = 13.4°

R1= 43.4°

R2 = 136.6°

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 triangle 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?


Students may find it easier to use the NonEuclid website 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 congruent 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 triangles in the hyperbolic plane are congruent. This point will

be addressed in questions that are posed ahead.

31

1

3 m1 = 25.7°

m2 = 7.0°

m3 = 56.0°

m4 = 25.7°

m5 = 7.0°

m6 = 133.0° 6

45

2

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?


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 .

32

l1 l2l1 = 3.59 l2 = 3.59

12

m1 = 15.5°

m2 = 15.5°

B C

A

In the hyperbolic case, the NonEuclid program once again offers students 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 valid 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.


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

and therefore .

As in Euclidean geometry, if two angles of a triangle on the hyperbolic plane are

congruent, then the sides opposite the angles are congruent.

33

B C

A

24. a) Draw an equilateral triangle on the Euclidean plane. Prove that the measure

of each angle of an equilateral triangle is 600

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?

34

Distance = 3.12 Distance = 3.12

Distance = 3.12

1

2 3

m1 = 23.3°

m2 = 23.3°

m3 = 23.3°

m2= 17.3° m1 = 17.3°

Distance = 3.73

Distance = 3.73

12


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 method 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.

35

A B

C'

C

Students will discover that equilateral triangles on the hyperbolic plane are also

equiangular. Unlike Euclidean equilateral triangles, however, each angle is not

equal to 60o, 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.

36

AB

C

C'

A B

C

h

b


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. Does 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 command under the measure command to discover

whether this theorem is valid on the hyperbolic plane.

37

b1

b2b3

b1 = 3.58

h1

h2 = 2.40 b2 = 4.05

h2

0.5 h2 b2 = 4.87

h3

h3 = 2.51 b3 = 3.94

h1 = 2.87

0.5 b1 h1 = 5.13

0.5 h3 b3 = 4.95


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 drawn 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 are 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 create 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).

[

38

B

C

A

1

m1 = 90.0°

a = 2.78

b = 2.12

c = 4.22

a

b

c

a2 + b2 = 12.21

c2 = 17.83

27. The SAS Congruence 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 the hyperbolic plane.


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

construct congruent to in another 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.

39

A

C

B D

F

E

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


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?

40

A

C

B D

F

E


Construct 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 congruent. Similar triangles do not exist on the

hyperbolic plane unless the two triangles are congruent; in which case they are

identical.

41

A

C

B

D

E

F A = 22.3° B = 53.5°

D = 22.3°

E = 53.6°

C = 34.3°

F = 73.9°

module for hyperbolic geometry

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