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.
2.
Line segment:
Circle :
The segment AB, AB , consists of the points A and B and
all the points on line AB that are between A and B
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 180 o
.
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

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
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.
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. n m l
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 xaxis. l m
P n
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

A A
C
C
B
B x
Finding angle measure in the Poincaré Half Plane model x
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.
C m
Lines in the Poincaré model
7
A
B
Constructing the angle between two lines in the Poincaré model.

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 Sketchpad 1
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

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

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.
C D
A
B
10

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
A line in the Poincaré disk becomes straight in the Euclidean sense when it passes through the center of the 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 center of the disk and as it becomes closer and closer to the center, it becomes more straight in the
Euclidean sense.
11

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?
Q
P
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

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.
A C B b) Does the Betweenness Axiom hold on the hyperbolic plane?
Note to the teacher: (5 minutes)
The Betweenness Axiom holds for hyperbolic plane.
P
Q
R
P Q
R
5.
a) Draw two lines on the Euclidean plane. In how many points do these lines intersect?
13

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

P m
Note to the teacher. (15 minutes)
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?
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 same be said of lines in the hyperbolic plane?
15

PQ = 1.94
PR = 2.56
PS = 3.18
PT = 5.33
ST = 3.38
Q
R
P S
T
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 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

Distance = 1.91
Distance = 1.91
Distance = 1.91
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 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.
C
A
E
D
B 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

R
P
2
1
O
4
3
S
Q m1 = 70.0° m2 = 70.0° m3 = 110.0° m4 = 110.0°
Note to the teacher: (15 minutes)
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?
A
18

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.
P
1 m1 = 90.0°
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 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 .
P l
M
Q
1
N m1 = 90
19

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. a a 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.
T
R
P
Note to the teacher: (10 minutes)
V
107.8
S
W
84.4
Q
U
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

13.
a) Draw a pair of parallel lines on the Euclidean plane. Prove that the alternate interior angles are congruent. a a b) Draw a pair of parallel lines on the hyperbolic plane. Measure the alternate angles and determine whether they are congruent.
R
P
T
V
107.8
S
84.4
W
U
Q
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. a b
21

b) Draw a pair of parallel lines on the hyperbolic plane. Measure the consecutive interior angles. Are these pairs of angles supplementary?
R
T
V
107.8
S
P
95.6
W
U
Q
Note to the teacher: (10 minutes)
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

2
1 m l m1 = 90.0° m2 = 35.9°
Note to the teacher: (10 minutes)
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?
B
A n m l
23

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? t
Q s
P P
Q s t r r
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-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

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?
R
P
Q m 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 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

C l'
1
A'
2
2
A
1 l B
Consider two lines l and l’
and a transversal as shown in the figure, and let

A
1
and

A
1
' 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 '

AB . Then we see that '

' ( ) . In particular,
  
A
2
. Therefore, since

A
1
' and

A '
2
are supplementary,

A
1
and

A
2 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

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.
3
1 m m1 = 90.0° m2 = 90.0°
2 m3 = 90.0°
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

A
1 and

A
1
' are corresponding angles.

A '
2
and

A
1
' are vertical angles that are congruent in both geometries. Therefore, if
  
A
1
, then
  
A
1
. These two are alternate interior angles and by the Alternate Interior Angle Theorem we conclude that l and l'
’are parallel.
A'
2
1 l'
A
1 l
27

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
A l
1
2
3
C m 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.
Q
P
2
1
3
R m1 = 17.7° m2 = 12.9° m3 = 11.6° m1 + m2 + m3 = 42.2°
28

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.

C
2
and

B are alternate interior angles, and therefore congruent,

C
3
and

A are corresponding angles and are therefore congruent, and the sum of the three angles

C
1
,

C
2
and

C
3
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 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.
A
B
1 2
C D
29

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?
P
Q
1
R
2
S
P=14.2°
Q = 13.4°
R1= 43.4°
R2 = 136.6°
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
,
     
1
180 o
. But the adjacent angles at
C are supplementary, so C
1
C
2
180 o
. Therefore,

C
2
  
B
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 R
1
R
2
180 o
and
R
1
P Q is less than 180 o , then

R
2
is more than the sum of

P and

Q . 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.
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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?
1
2
3
5
6
4 m1 = 25.7° m4 = 25.7° m2 = 7.0° m5 = 7.0° m3 = 56.0° m6 = 133.0°
Note to the teacher. (15 minutes)
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.
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22.
a) Draw an isosceles triangle on the Euclidean plane. Prove that the angles at the base of the congruent sides are congruent.
B
A
C b) Draw an isosceles triangle on the hyperbolic plane. Measure the angles at the base of the congruent sides. Are the base angles congruent? l1 = 3.59 l1
1 m1 = 15.5°
Note to the teacher: (15 minutes) l2 l2 = 3.59
2 m2 = 15.5°
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
AB

,

,

BC . Corresponding angles are therefore congruent, and B C .
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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
A
C 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 ABC

ACB using ASA, and therefore AB

AC .
As in Euclidean geometry, if two angles of a triangle on the hyperbolic plane are congruent, then the sides opposite the angles are congruent.
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Distance = 3.73
1 m1 = 17.3°
Distance = 3.73
2 m2= 17.3°
24.
a) Draw an equilateral triangle on the Euclidean plane. Prove that the measure of each angle of an equilateral 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?
Distance = 3.12
2
1
Distance = 3.12
Distance = 3.12
3 m1 = 23.3° m2 = 23.3° m3 = 23.3°
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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 AB 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.
C
A B
C'
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.
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C
A
C'
B
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?
C
A b h
B b) Investigate whether this formula is valid for calculating the area of a triangle on the hyperbolic plane.
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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. b2 h2 h1 h3 b3 b1 = 3.58 h1 = 2.87
0.5 b1 h1 = 5.13
h2 = 2.40 b2 = 4.05
0.5 h2 b2 = 4.87
b1 h3 = 2.51 b3 = 3.94
0.5 h3 b3 = 4.95
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.
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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.
B a c m1 = 90.0° a = 2.78 b = 2.12
C
1 b
A c = 4.22 a 2
+ b
2
= 12.21
c 2
= 17.83
Note to the teacher:
In general, when we say that two triangles ABC 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).
[
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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.
Note to the teacher: (20 minutes)
F
C
E
A B D
Students will construct ABC on the disk. Then, using the tool measure, they will construct DE congruent to AB 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 AC

DF .
We notice that these two triangles have two sides and an angle between them congruent ( AC

DF ,
  
,

DE ) . Now what we need to do is to check that all other corresponding components are congruent as well
(i.e. BC

EF ,
  
F ,
  
E ). Students will find through measurement that all other corresponding components are congruent.
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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
C
A B D
F
E
Note to the teacher: (20 minutes)
Once again, students will construct ABC on the disk. Using the tool measure, construct DE

,

,

BC . 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?
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Note to the teacher: (20 minutes)
C
D
A B
F
E
A = 22.3°
B = 53.5°
C = 34.3°
D = 22.3°
E = 53.6°
F = 73.9°
Construct ABC 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

A . At vertex E construct an angle that is congruent to the angle at B. The student will discover the third angle of DEF is only congruent to the third angle of ABC 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.
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