Download Non-Euclidean Geometry Unit

Non-Euclidean Geometry Unit
Euclidean Geometry: The geometry with which we are most familiar is called Euclidean
geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300
BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about
squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems
which are taught in high schools today can be found in Euclid's 2000 year old book.
Euclidean geometry is of great practical value. It has been used by the ancient Greeks through
modern society to design buildings, predict the location of moving objects and survey land.
Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from
Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions,
assumptions, and proofs that describe such objects as points, lines and planes. The two most
common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The
essential difference between Euclidean geometry and these two non-Euclidean geometries is the
nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line
through the point that is in the same plane as the given line and never intersects it. In spherical
geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that
pass through the point and are parallel to (in the same plane as and do not intersect) the given
line.
Euclid’s First Five Postulates:
1. A unique straight line can be drawn through any two points A and B
2. A segment can be extended indefinitely
3. For any two distinct points A and B, a circle can be drawn with center A and radius AB
4. All right angles are congruent
5. Given a line L and a point P not on L, there exists a unique line though P parallel to L.
Euclidean Geometry is Geometry in which the parallel postulate is true.
Spherical Geometry: Spherical geometry is a plane geometry on the surface of a sphere. In a
plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined
in the usual way, but lines are defined such that the shortest distance between two points lies
along them. Therefore, lines in spherical geometry are Great Circles. A Great Circle is the largest
circle that can be drawn on a sphere. The longitude lines and the equator are Great Circles of the
Earth. Latitude lines, except for the equator, are not Great Circles. Great Circles are lines that
divide a sphere into two equal hemispheres.
Spherical geometry is used by pilots and ship captains as they navigate around the globe.
Working in spherical geometry has some non-intuitive results. For example, did you know that
the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The
Philippines are south of Florida - why is flying north to Alaska a short-cut? The answer is that
Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a
Great Circle). Another odd property of spherical geometry is that the sum of the angles of a
triangle is always greater then 180°. Small triangles, like those drawn on a football field, have
very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York,
L.A. and Tampa) have significantly more than 180°.
Hyperbolic Geometry: Hyperbolic geometry is a "curved" space, and plays an important role in
Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within
the field of Topology.
Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a
novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also
has many differences from Euclidean geometry.
Here are the websites that we will be using for this unit:
Spherical Geometry
 Go to: http://www.plu.edu/~heathdj/java/
 Then click on “geometry playground”
 That will open up an applet
 There are several models. Use the “spherical” model.
Hyperbolic Geometry.
 Go to: http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html
 Scroll to the middle of the page and click on “Run NonEuclid using Java Web Start
WITHOUT Save and Print Permissions”
Spherical Geometry Unit
Spherical Geometry is geometry in which Euclid’s fifth postulate is replaced with the following:
Given a line G and a point P not on G, every line through P intersects G; that is, no line through
P is parallel to G.
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.
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.
Circle: The set of all points, P, in a plane that are a fixed distance from a fixed point, O, on that
plane, called the center of the circle.
Parallel lines: Two lines, l and m on the plane are parallel if they do not intersect.
Sphere: The set of the points in space that are a given distance from a fixed point, called the
center of the sphere.
Great Circle: A Great Circle is a circle whose center is the center of the sphere and whose radius
is equal to the radius of the sphere. The Great Circle in spherical geometry is a line.
Arc of a Great Circle: The shortest path between two points on the sphere is the arc of a Great
Circle.
Antipodal points (Pole Points): Points that lie at the intersection of a Great Circle and a line
through the center of the circle on the sphere.
Small Triangle: The small triangle is formed by joining three non-collinear vertices with the
shorter arc between the vertices. Three vertices then determine only one spherical triangle.
Questions
Answer the following underlined questions on an answer sheet. Many of the answers have
clues by the diagrams given in the problem but you should confirm the findings by drawing
your own figures on the Lenart sphere (starting with question 5). Generally, when the
problem is divided into parts “a” and “b”, the “a” part asks about what we currently know
in Euclidean (Plane) Geometry.
Definition of a sphere
1) Spherical geometry is geometry on a sphere. Define a sphere.
Note: In spherical geometry you will be working on the surface of the sphere and not in the
interior of the sphere
Shortest Path
2a) Locate two points in the plane and label them P and Q.
What is the shortest path between two points on a plane?
2b) Locate two points on the sphere and label them A and B. (Do not locate these points such
that they are opposite each other on the sphere. Such opposite points are called antipodal points
and they will be referred to later on in the activity.)
What is the shortest distance between two points on a sphere?
Segments
3a) If segment PQ (from problem 2a) in the plane were extended
indefinitely beyond the graph, how far would it go?
3b) If you extended segment AB on your sphere, describe what the result
would be.
3c) You have just drawn a Great Circle. Define a Great Circle and name a
Great Circle on Earth.
3d) Latitudes are the horizontals on the earth that determine north from south, (see picture
above). Are all lines of latitude Great Circles?
3e) Longitudes are verticals on the earth that connect the North Pole with the South Pole and that
determine east from west, (see picture above). Are all lines of longitudes Great Circles?
3f) In the Euclidean plane the shortest path from P to Q is unique, and its measure is fixed. Can
the same be said of the segment AB on the sphere? Is the measure of a segment AB on a sphere
unique?
Measuring segments
To measure the distance between two points P and Q in the Euclidean plane, you would use a
ruler or perhaps the distance formula. The units of measurement would be a linear standard unit
of measurement (i.e. inches, centimeters, miles, etc.).
In spherical geometry the distance between two points is measured in degrees, that is a fraction
of the Great Circle which contains the segment that connects the two points.
In this problem we will discuss how distances are measured on a sphere. Suppose the Earth is a
sphere. In Euclidean space the Earth has a radius of 6,400 km (the radius in this case as
measured from the center of the sphere to any point on the surface of Earth is 6,400 km).
4a) What is Earth’s circumference?
4b) How many degrees does this represent?
4c) If two places on Earth are opposite each other (i.e. poles or antipodal points), what is the
distance between them in kilometers? In degrees?
4d) If two places are 90o apart from each other, how far apart are they in kilometers?
4e) If two places are 5026 km apart, what is their distance apart measured in degrees (nearest
degree?
4f) Mars has a circumference of 21,320 kilometers. What does this distance represent in degrees?
4g) What is the furthest distance that two places on Mars can be apart from each other in
degrees? In kilometers (in the spherical sense)?
In the following problems, use the Lenart sphere to investigate your results
Euclid’s First Postulate
5) Euclid’s first postulate states that for every point P and every point Q, where P is not equal to
Q, there exists a unique line l through P and Q. On the Lenart sphere, draw two points and
connect them with the shortest path possible. Is Euclid’s first postulate valid in spherical
geometry?
P
P
Q
Q
6a) In how many points can two lines on a plane intersect?
6b) Use your sphere to draw two Great Circles on the sphere. In how many points can two lines
on the sphere intersect?
B
Remember, in spherical geometry Great Circle = Line
7) Euclid’s Fifth Postulate (Parallel Postulate):
A
Euclid’s fifth postulate state: “Given a line L and a point P not on L, there exists a unique line
though P parallel to L.” We know that the definition of parallelism is lines that do not intersect.
Do parallel lines exist in spherical geometry?
How would you re-word Euclid’s Fifth Postulate so that it is true for spherical geometry?
Betweenness
8) Locate a point R between two points P and Q on the plane.
A
C
P
R
B
Q
The Betweenness Axiom states that if P, Q, and R are three points in the plane, then one and only
one point is between the other two.
Draw a Great Circle on your sphere and locate a point C between points A and B on the sphere.
Is the Betweenness Axiom valid for the three points that are drawn on the sphere?
Euclid’s Second Postulate
9) Euclid’s second postulate states: “a line segment can be extended
infinitely from each side.” Is this postulate valid in spherical geometry?
Please explain.
Circle on a sphere
10) A circle is defined as the set of points equidistant from a given point
Draw a circle on your sphere using your compass set at 45°. Label the two
centers of the circle (yes, there are two centers). The first radii is 45°.
Measure the second radii. How are the radii lengths related to each other?
Euclid’s Third Postulate
11) Euclid’s third postulate states, “A circle can be drawn with any center and any radius. Is this
true for circles on the sphere?
P
S
O
Vertical Angles
R
Q
12) We know that when two lines intersect on the plane, the measures of the vertical angles are
congruent. We can confirm that using a protractor.
Draw two Great Circles on your sphere. Label the points of intersection A and B. Use your
spherical protractor to measure the pairs of vertical angles formed at the point of intersection of
the Great Circles.
B
What do you notice about each pair of vertical angles?
A
Perpendiculars
Draw a Euclidean line. Locate a point P that is not on the line. What is the shortest path from the
point to the line? This path is called the distance from P to the line. Construct this path.
P
13) Draw Great Circle on the sphere. Locate a point P that is not on the Great Circle and not a
pole point. What is the distance (shortest path) from the point to the arc?
How many perpendiculars can be drawn from the point P to the arc?
Locate a pole point for this Great Circle. How many perpendiculars can be drawn from this pole
to the Great Circle?
P
14a) Draw two intersecting lines l and m on the plane. Can you draw a common perpendicular
to these two lines?
Draw two parallel lines l and m on the plane. Can you draw a common perpendicular to these
two lines? If you can, how many can be drawn?
14b) Draw two great circles that are not perpendicular. Can a common perpendicular be drawn
to these two Great Circles?
Three lines
15a) In how many different ways can three lines intersect in the plane (see picture)?
m
n
m
m
m
n
n
p
p
n
p
15b) In how many points do two geodesics lines (Great Circles) on a sphere intersect (picture
1)? In how many ways do three geodesics lines (Great Circles) on the sphere intersect (picture
2)? Note, you may want to construct these circle on your sphere.
Picture 1
Picture 2
Lune
16a) What is the minimum number of sides required to draw a closed figure in the plane using
straight lines only?
Name the figure you drew in the plane.
16b) What is the minimum number of sides required to draw a closed figure on
the sphere?
You may have decided that the term biangle would be appropriate for this shape.
Another name for this figure is a lune. How many lunes are formed by the
intersection of two Great Circles?
What is the relationship between the two points of intersection of the sides of the lune?
How long are the sides of the lunes?
Measure the opposite angles of the lune. What do you notice?
p
Triangles
17a) Construct three non-collinear points in the plane. Connect them
to form a triangle. How many triangles can you form?
17b) Locate three non-collinear points A, B, and C, on the sphere. Draw Great Circles through
AB, AC, and BC.
How many different triangles with vertices A, B, and C, can be drawn?
(Use of different colors may help to identify the triangles more easily.)
B
A
We will define the triangle formed using the shorter arcs joining two points on the sphere as the
“small triangle.” Identify and shade in the small triangle on the sphere.
18a) What is the sum of the interior angles of the triangle?
18b) Measure the angles of the small triangle ABC. What is the sum of the measures of the
angles of the small spherical triangle?
Draw another larger triangle and measure its angles and find the angle sum. Is the angle sum the
same for both triangles?
C
Triangle Sum
19a) Is it possible for a triangle on the plane to have more than one right angle?
19b) Is it possible for a triangle on the sphere to have more than one right angle?
Triangle Sum Theorem
The sum of the interior angles in spherical triangle is greater than ______ and less than _______.
Exterior Angles of a Triangle
20a) State the exterior angles theorem for triangles on a plane.
20b) Draw ∆ABC on the sphere. Extend BC to D and measure  ACD. Find the measure of
both  A and  B. Is there a relationship between the exterior angle of a triangle on the sphere
and the non-adjacent interior angles?
Third Angles Theorem
21a) Suppose two angles of one triangle are congruent to two angles of another triangle on a
plane. How does the measure of the third angles of the triangles compare?
21b) Draw ∆ABC on the sphere. Measure the size of each angle of the triangle. Construct a
second ∆DEF with A  D and B  E and DE = 2AB and FE = 2BC. Measure the third
angle of the triangle and compare this measure with the measure of the third angle of triangle
ABC. [Note, you will use these two triangle to answer question 22b as well].
In the plane, the Third Angles Theorem states that if two angles of one triangle are congruent to
two angles of another, then the third angles are congruent. Does this theorem apply to triangles
on the sphere?
Similar Triangles
22a) Given the diagram, how do the measures of AC and DF compare?
D
A
x
B
2x
y
C
F
2y
22b) Given the two triangles drawn on the sphere in problem 21A, measure AC and DF. How do
the measures of AC and DF compare?
E
In the plane, two triangles are said to be similar if all of their corresponding angles are congruent
and all of their corresponding sides are proportional. Does this theorem apply to triangles on the
sphere?
22c) Draw a triangle on the sphere. Measure the angles of the triangle. Construct a second
triangle with angles equal in size to the angles of the first triangle. Now measure the sides of
both triangles. What do you notice? Summarize your findings in terms related to your study of
plain geometry.
Pythagorean Theorem
23a) State the Pythagorean Theorem with respect a right triangle in the plane.
23b) Investigate whether this theorem is relevant on the sphere. We have already discovered that
it is possible to draw a triangle on the sphere with one, two, or three right triangles. Construct
one of each of these triangles on the sphere and investigate whether there is any relationship
between the sides of the triangle. Does the Pythagorean Theorem hold in Spherical Geometry?
SSS Triangle Congruency
24a) In plane geometry, if three sides of one triangle are congruent to three sides of a second
triangle, then are the two triangles congruent?
24b) Construct a triangle on the sphere. Measure the length of the sides of the triangle. Construct
a second triangle with sides equal in measure to the sides of the first triangle. Then measure the
angles of the two triangles. Are the two triangles congruent?
SAS Triangle Congruency
25a) In plane geometry, if two sides and the included angle of one triangle are congruent to two
sides and the included angle of a second triangle, then are the two triangles congruent?
25b) Draw ∆ABC on the sphere. Measure the lengths of the sides AB and BC and the measure
of  DEF where the measure of AB = DE, BC = EF and ABC  DEF. Are the two triangles
congruent?
ASA Triangle Congruency
26a) In plane geometry, if two angles and the included side of one triangle are congruent to two
angles and the included side of a second triangle, then are the two triangles congruent?
26b) Construct ∆ABC on the sphere. Measure the length of BC and the measure of  B and 
C. Construct ∆DEF with the measure of BC = EF and B  E and C  F . Are the two
triangles congruent?
Area of a Sphere and Lune
27) The formula for the surface area of a sphere is 4π r2 where r is the radius of the sphere in the
Euclidean sense (the distance from the interior center of the sphere to any point on the surface of
the sphere).
What is the area of this lune with an interior angle of 60°?
What is the area of this lune with an interior angle of 90°?
Write down a generalized formula for the area of a lune.
Finding the area of a spherical triangle (Girard’s Theorem)
28a) Find area of a triangle on the plane.
28b) We will now derive a formula for the area of a triangle on the sphere. In order to
understand how the formula is derived, you are encouraged to draw triangles on the sphere and
use colors to identify the different triangles under discussion. This is very important to a clear
understanding of the derivation of the formula for the area of a triangle on the sphere. This
formula is commonly known as Girard’s Theorem.
Draw a triangle on the sphere and label the angles α, β, γ (alpha, beta, gamma) as shown in the
diagram below
Using colors, draw and shade the α-lunes. Notice that there is a congruent α-lune on the back of
the sphere. Repeat this for the two β-lunes and the two γ-lunes using different colors. Notice
that the triangle ABC appears in each of the lunes.
Notice also, that there is a copy of triangle ABC in each of the lunes on the back of the sphere.
Also notice that you shaded in both triangle ABC and the copy of ABC with all three colors.
If we now wished to get an expression for the area of the sphere in terms of the area of the lunes,
we would get the following (luneα = area of lune α)
Area of sphere = 2luneα + 2luneβ + 2 luneγ - 4ΔABC



)4 r 2  2(
)4 r 2  2(
)4 r 2  4ABC
360
360
360
Switch triangle and sphere formulas: 4ABC  2(  )4 r 2  2(  )4 r 2  2(  )4 r 2  4 r 2
360
360
360



4ABC  (
)4 r 2  (
)4 r 2  (
)4 r 2  4  r 2
Reduce (2/360) to (1/180):
180
180
180



ABC  (
) r 2  (
) r 2  (
) r 2   r 2
Divide all terms by 4:
180
180
180
      180
Factor out π r2 and get a common
ABC   r 2 (
)
180
denominator:
Substitute area of sphere and lune:
4 r 2  2(
This formula has a very interesting consequence for the area of a triangle on the sphere. It states
that the area of a triangle on the sphere is directly related to the angles of the triangle.
      180
) is called Girard's Theorem, and the
180
quantity α + β + γ - 180o is called the spherical excess of the triangle
Once again the formula ABC   r 2 (
Calculate the area of a spherical triangle with angles 90o, 90o, and 90o.
Draw these triangles on the sphere and confirm that the answer you got for the area is consistent
with what you would have expected starting with the formula for the sphere.
A = 4πr2
Calculate the area of a spherical triangle with angles 45o, 45o, and 145o.
Advanced Study: Area of a n-sided polygon (n > 3)
29) Investigate the area of a quadrilateral using the diagram below, then determine a formula for
calculating the area of an n-sided polygon.
C
1 3
B
D
2 4
A
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Hyperbolic Geometry
Hyperbolic Geometry is geometry in which Euclid’s fifth postulate is replaced with the
following:
Hyperbolic Parallel Postulate:
Given a line l, and a point A, not lying on l, there exists at least two lines through A that are
parallel to line l.
Hyperbolic geometry takes place on a curved 2-dimensional surface called hyperbolic space.
In hyperbolic space, every point looks like a saddle.
Terms used in the modules will be defined as follows:
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
Circle: The set of all points, P, that are a fixed distance from a fixed point, O, called the center of
the circle.
Parallel lines: Two lines, l and m are parallel if they do not intersect.
Polygons: A sequence of points and geodesic segments joining those points
Angle Measures: The measure of an angle is the radian measure of the angle formed by
 The tangent rays at a point of intersection of two arcs, or
 An ordinary ray and the tangent ray at a point of intersection of the arc and the ordinary
ray
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 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 Poincaré Half Plane Model for Studying Hyperbolic Geometry.

The Poincaré Disk Model for Studying Hyperbolic Geometry. (This is the model we will
be using)
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
A
B
Lines in the Poincaré model
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.
Questions
Answer the following underlined questions on the answer sheet. Many of the answers are
evident by the diagrams given in the problem (which should guide your lab), but you
should confirm the findings by drawing your own figures on the hyperbolic geometry
software. When the problem is divided into parts “a” and “b”, the “a” part asks about
what we currently know in Euclidean (Plane) Geometry. You should be able to answer the
“a” part without investigation.
1) What is the shortest path between two points in the Euclidean Plane?
2) 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 (File-New). 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
3) Consider 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
Does the Betweenness Axiom hold on the hyperbolic plane?
P
P
Q
R
Q
R
4a) Draw two lines on the Euclidean plane. In how many ways do these lines intersect?
4b) Create a new disk (File-New). Draw two lines on the hyperbolic plane. In how many points
do the lines intersect?
5a) Two lines are defined as being parallel if they have no points in common. Given a line l on
the Euclidean plane and a point A that is not on l. How many possible lines can you construct
through A that is parallel to l?
A
l
5b) Create a new disk (File-New). 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?
P
m
6) How would you re-word Euclid’s parallel postulate so that it is true for the hyperbolic plane?
7) Lines in Euclidean geometry are of infinite length. Can the same be said of lines in the
hyperbolic plane? Explain.
PQ = 1.94
PR = 2.56
PS = 3.18
PT = 5.33
Q
P
ST = 3.38
R
S
T
8) Euclid’s third postulate states that a circle can be drawn with any center and any radius.
Create a new disk (File-New). 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? Explain.
Distance = 1.91
Distance = 1.91
Distance = 1.91
9) Vertical angles on the Euclidean plane are congruent.
Create a new disk (File-New). Draw two lines on the hyperbolic
plane.
 Measure the pairs of adjacent angles. Are they
supplementary?
 Measure the vertical angles. Are the pairs of vertical angles
congruent?
m1 = 70.0°
R
m2 = 70.0°
1
2
O
4
Q
m4 = 110.0°
3
P
S
10) On Euclidean plane, given a line l and a point A not on the line, only one perpendicular can
be drawn to line l through point A.
Create a new disk (File-New). 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
m3 = 110.0°
m1 = 90.0°
11) On Euclidean plane, given a pair of parallel lines and a transversal, each pair of
corresponding angles are congruent.


Create a new disk (File-New). Draw a pair of parallel lines and a transversal on the hyperbolic
plane. Measure the pairs of corresponding angles. Is the corresponding angles postulate valid on
the hyperbolic plane?
T
V
S
107.8
R
W
84.4
P
Q
U
12) On Euclidean plane, 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.
Create a new disk (File-New). Draw two parallel lines l and m on the hyperbolic plane. At a
point on l draw a perpendicular transversal. Is the Perpendicular Transversal Theorem valid on
the hyperbolic plane?
1
m
m1 = 90.0°
l
2
m2 = 35.9°
13) On Euclidean plane, if two lines are parallel to the same line, then they are parallel to each
other.
n
B
m
A
l
Create a new disk (File-New). 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?
Q
t
Q
s
s
P
P
t
r
r
14) On Euclidean plane, if two lines are perpendicular to the same line, then they are parallel to
each other.
Create a new disk (File-New). 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?
Q
P
R
m
15) We know that the sum of the measures of the angles of a triangle equals 180° on the
Euclidean plane.
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Use the hyperbolic
measure tool to measure the interior angles of the triangle. Does the Triangle Sum Theorem hold
on the hyperbolic plane?
P
1
m1 = 17.7°
3
m2 = 12.9°
R
2
m3 = 11.6°
Q
m1 + m2 + m3 = 42.2°
16) We know that an exterior angle of a triangle equals the sum of its two non-adjacent interior
angles in the Euclidean plane.
Create a new disk (File-New). 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. Does the exterior angle theorem
hold on the hyperbolic plane?
P
P=14.2°
R
Q
Q = 13.4°
2
1
S
R1= 43.4°
R2 = 136.6°
17) On the Euclidean plane, if two angles of one triangle are congruent to two angles of another
triangle, then the third angles are congruent.
Create a new disk (File-New). 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
3
m1 = 25.7° m4 = 25.7°
m2 = 7.0° m5 = 7.0°
2
6
5
m3 = 56.0° m6 = 133.0°
4
18) On Euclidean plane, the base angles of an isosceles triangle are congruent.
Create a new disk (File-New). Draw an isosceles triangle on the hyperbolic plane. Measure the
angles at the base of the congruent sides. Are the base angles congruent?
A
l1 = 3.59 l1
l2 l2 = 3.59
2
1
m1 = 15.5°
B
C
m2 = 15.5°
19) On Euclidean plane, if a triangle has two angles congruent then the sides opposite he
congruent angles are congruent.
Create a new disk (File-New). Draw a triangle with two angles congruent on the hyperbolic
plane. Measure the sides opposite the congruent. Are those sides congruent?
A
Distance = 3.73
Distance = 3.73
2
1
m2= 17.3°
m1 = 17.3°
B
C
20) On Euclidean plane, each measure of each angle of an equilateral triangle is 60°
Create a new disk (File-New). 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?
1
m1 = 23.3°
Distance = 3.12
Distance = 3.12
m2 = 23.3°
m3 = 23.3°
3
2
Distance = 3.12
21) In a right triangle on the Euclidean plane, the square of the hypotenuse is equal to the sum of
the squares on the legs of the triangle. Does Pythagorean Theorem 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.
B
m1 = 90.0°
a = 2.78
a
b = 2.12
c
c = 4.22
C
a 2 + b 2 = 12.21
1
c 2 = 17.83
b
A
22) 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.
F
C
E
A
B
D
23) The SSS Congruence Postulate states that if three sides of one triangle are congruent to three
sides of another triangle, then the two triangles are congruent.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your
findings.
F
C
E
A
B
D
24) The ASA Congruence Postulate states that if two angles and the included side of one triangle
are congruent respectively to two angles and the included side of another triangle, then the two
triangles are congruent.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your
findings.
F
C
E
A
D
B
25) 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.
On the hyperbolic plane, if three angles of one triangle are congruent to three angles of another,
what can we say about these two triangles?
A = 22.3°
F
C
B = 53.5°
E
C = 34.3°
D = 22.3°
D
E = 53.6°
F = 73.9°
A
B
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Spherical Geometry
Worksheet Answer Form
Name ______________________________
Period ________ Date _________________
1.
Define a sphere:___________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
2a. What is the shortest path between two points on a plane? :__________________________
______________________________________________________________________________
______________________________________________________________________________
2b. What is the shortest path between two points on a sphere? :_________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
3a. How far would a segment go if extended indefinitely on a plane? _______________
______________________________________________________________________________
______________________________________________________________________________
3b. How far would a segment go if extended indefinitely on a sphere? _______________
______________________________________________________________________________
______________________________________________________________________________
Define a great circle. :______________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Name a great circle on Earth:_________________________________________________
______________________________________________________________________________
Are all lines of latitude great circles? Explain your answer. _______________________
______________________________________________________________________________
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______________________________________________________________________________
______________________________________________________________________________
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Is the measure of a segment AB unique on a sphere? Explain your answer. ____________
______________________________________________________________________________
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4.
Could you use a ruler or the distance formula to measure the distance between A and B on
the sphere? ____________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5a.
What is the Earth’s circumference? ____________________________________________
5b.
How many degrees does this represent? ________________________________________
5c.
If two places on Earth are opposite each other, what is the distance between them in
kilometers in the spherical sense? ________________ In degrees? ________________
5d.
If two places are 90° apart from each other, how far apart are they in kilometers in the
spherical sense? ________________________________________________________
5e.
If two places are 5026 km apart, what is their distance apart measured in degrees? ______
5f.
What does this distance represent in degrees? ___________________________________
5g.
What is the furthest distance that two places on Mars can be apart from each other in
degrees? _______________ In kilometers (in the spherical sense)? _______________
Is Euclid’s first postulate (for every point P and every point Q, where P is not equal to Q,
there exists a unique line l through P and Q) valid in spherical geometry? Justify your
answer. _______________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6.
7a.
In how many points do two lines on a plane intersect? _____________________________
7b. In how many points do two lines on the sphere intersect? __________________________
______________________________________________________________________________
How would you re-word Euclid’s parallel postulate so that it is true for spherical
geometry?_____________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
8.
9.
Is the betweenness Axiom valid for the three points that are drawn on the sphere? Justify
your answer. __________________________________________________________
______________________________________________________________________________
Is Euclid’s second postulate (a line segment can be extended infinitely from each side)
valid in spherical geometry? Justify your answer. _____________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
10.
11a. Define the term circle. ______________________________________________________
______________________________________________________________________________
______________________________________________________________________________
11b. How are the radii lengths of a circle drawn on a sphere related to each other? __________
______________________________________________________________________________
______________________________________________________________________________
Is Euclid’s third postulate (that a circle can be drawn with any center and any radius) true
for a sphere? __________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
12.
13. What is the relationship of two vertical angles drawn on a sphere? ___________________
______________________________________________________________________________
______________________________________________________________________________
14. What is the distance (shortest path) from the point to the arc? _______________________
______________________________________________________________________________
How many perpendiculars can be drawn from the point P to the arc? _________________
______________________________________________________________________________
How many perpendiculars can be drawn from a pole point to the great circle? __________
______________________________________________________________________________
15a. Can you draw a common perpendicular to two intersecting lines on a plane? ___________
Can you draw a common perpendicular to two parallel lines? If you can, how many can
be drawn? ____________________________________________________________
15b. Can a common perpendicular be drawn to two great circles? ________________________
______________________________________________________________________________
16a. In how many different ways can three lines intersect in the plane? ___________________
16b. In how many points do two geodesics lines on a sphere intersect? ___________________
In how many ways do three geodesics lines on the sphere intersect? __________________
17a. What is the minimum number of sides required to draw a closed figure in the plane using
straight lines only? ______________________________________________________
Name the figure you drew in the plane. ________________________________________
17b. What is the minimum number of sides required to draw a closed figure on the sphere?
______________________________________________________________________________
How many lunes are formed by the intersection of two great circles? _________________
______________________________________________________________________________
What is the relationship between the two points of intersection of the sides of the lune?
______________________________________________________________________________
______________________________________________________________________________
How long are the sides of the lune? ___________________________________________
______________________________________________________________________________
What do you notice about the opposite angles of a lune? ___________________________
______________________________________________________________________________
______________________________________________________________________________
18a. How many triangles can you form using three points on a plane? ____________________
18b. How many different triangles with vertices A, B, and C, can be drawn on a sphere?
______________________________________________________________________________
19a. What is the sum of the interior angles of the triangle on a plane? ____________________
19b. What is the sum of the interior angles of the triangle on a sphere? ___________________
After drawing a second bigger triangle, is the angle sum the same for both triangles? ____
______________________________________________________________________________
20a. Is it possible for a triangle to have more than one right angle? _______________________
20b. Is it possible for a triangle to have more than one right angle? _______________________
21a. What is the relationship between a exterior angle of a triangle and its two non-adjacent
interior angles? ________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
21b. Is there a relationship between the exterior angle of a triangle and the non-adjacent interior
angles on the sphere? _________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
22a. If two triangle have two pairs of angles congruent, how does the measure of the third
angles of the triangles compare? __________________
How do the measure of PR and XZ compare? ___________________________________
22b. Does the Third Angles Theorem apply to triangles on the sphere? ___________________
______________________________________________________________________________
______________________________________________________________________________
23a. State the Pythagorean Theorem with respect to a right triangle in the plane. ___________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
23b. Does the Pythagorean Theorem hold in Spherical Geometry? _______________________
______________________________________________________________________________
______________________________________________________________________________
24a. What is the definition of similar polygons? _____________________________________
______________________________________________________________________________
______________________________________________________________________________
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What do you notice about the values of the ratios from the diagram? _________________
______________________________________________________________________________
24b. What do you notice about two triangles drawn on a sphere with angles equal in size?
______________________________________________________________________________
______________________________________________________________________________
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25a. Does SSS triangle congruence hold for triangles on a plane? ________________________
______________________________________________________________________________
25b. Does SSS triangle congruence hold for triangles on a sphere? _______________________
______________________________________________________________________________
26a. Does SAS triangle congruence hold for triangles on a plane? _______________________
______________________________________________________________________________
26b. Does SAS triangle congruence hold for triangles on a sphere? ______________________
______________________________________________________________________________
27a. Does ASA triangle congruence hold for triangles on a plane? _______________________
______________________________________________________________________________
27b. Does ASA triangle congruence hold for triangles on a sphere? ______________________
______________________________________________________________________________
28. Write down the formula for the surface area of a sphere. ___________________________
______________________________________________________________________________
What is the area of a lune with an angle of 60°? __________________________________
______________________________________________________________________________
What is the area of a lune with an angle of 90°? __________________________________
______________________________________________________________________________
Write down a generalized formula for the area of a lune. ___________________________
______________________________________________________________________________
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29a. Write down the formula for the area of a triangle in the Euclidean plane. ______________
______________________________________________________________________________
29b. Calculate the area of a spherical triangle with angles 90°, 90°, and 90°.________________
______________________________________________________________________________
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30. Determine a formula for calculating the area of an n-sided polygon. __________________
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Hyperbolic Geometry
Worksheet Answer Form
Name ______________________________
Period ________ Date _________________
1.
What is the shortest path between two points in the Euclidean Plane? _________________
______________________________________________________________________________
Does Euclid’s first postulate (that for every point P and for every point Q where P  Q, a
unique line passes through P and Q) hold in hyperbolic geometry? ________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
2.
3.
Does the Betweenness Axiom hold on the hyperbolic plane? _______________________
______________________________________________________________________________
______________________________________________________________________________
4a. In how many ways do two lines intersect on a plane? _____________________________
______________________________________________________________________________
4b. In how many ways do two lines intersect on a sphere? _____________________________
______________________________________________________________________________
______________________________________________________________________________
5a.
How many possible lines can you construct through point A that is parallel to line l on a
plane? ________________________________________________________________
5b.
How many possible lines can you construct through point P parallel through a line m in
the hyperbolic plane? Does the Parallel Postulate hold on the hyperbolic plane? _____
______________________________________________________________________________
______________________________________________________________________________
How would you re-word Euclid’s parallel postulate so that it is true for the hyperbolic
plane? ________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6.
7.
Are lines on the hyperbolic plane of infinite length? Explain. _______________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
8.
What appears to happen to a circle on the hyperbolic plane 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? Explain. ______________________
______________________________________________________________________________
______________________________________________________________________________
9.
Are two adjacent angles supplementary on the hyperbolic plane? ____________________
______________________________________________________________________________
Are the pairs of vertical angles congruent on the hyperbolic plane? __________________
______________________________________________________________________________
10. Can you construct a perpendicular from the point to the line on the hyperbolic plane? ___
______________________________________________________________________________
If so, how many perpendiculars can you construct? _______________________________
______________________________________________________________________________
11. Is the corresponding angles postulate valid on the hyperbolic plane? _________________
______________________________________________________________________________
12. Is the Perpendicular Transversal Theorem valid on the hyperbolic plane? _____________
______________________________________________________________________________
13.
On the hyperbolic plane, if two lines are parallel to the same line, must the original two
lines be parallel? _______________________________________________________
14.
On the hyperbolic plane, if two lines are perpendicular to the same line, must the original
two lines be parallel?
_______________________________________________________
15.
Does the Triangle Sum Theorem hold on the hyperbolic plane? _____________________
16.
Does the Exterior Angle Theorem hold on the hyperbolic plane? ____________________
17.
Does the Third Angles Theorem hold on the hyperbolic plane? ______________________
18.
Does the Base Angles Theorem hold on the hyperbolic plane? ______________________
19.
Does the Converse of the Base Angles Theorem hold on the hyperbolic plane? _________
20.
How do your observations of an equilateral triangle on the hyperbolic plane compare with
those on the Euclidean plane? _____________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
21. Does Pythagorean Theorem hold for triangles on the hyperbolic plane? _______________
______________________________________________________________________________
22. Does SAS triangle congruence hold for triangles on the hyperbolic plane? _____________
______________________________________________________________________________
23. Does SSS triangle congruence hold for triangles on the hyperbolic plane? _____________
______________________________________________________________________________
24. Does ASA triangle congruence hold for triangles on the hyperbolic plane? ____________
______________________________________________________________________________
25.
On the hyperbolic plane, if three angles of one triangle are congruent to three angles of
another, what can we say about these two triangles? ___________________________
______________________________________________________________________________
Non-Euclidean Geometry Study Guide:
1. What is the nature of parallel and perpendicular lines in Euclidean, Spherical,
and Hyperbolic geometry?
2. What is the nature of the sum of the interior angles of a triangle in Euclidean,
Spherical, and Hyperbolic geometry?
3. What is the concept of “betweenness” in Euclidean, Spherical, and Hyperbolic
geometry?
4. In Spherical Geometry, know the nature of the spherical plane, including Great
Circles (lines), points, segments, polygons and circles
5. Know the difference of properties of triangles between Euclidean, Spherical and
Hyperbolic geometry including Pythagorean Theorem, Exterior Angles
Theorem, and triangle similarity and congruency theorems.
6. In Spherical Geometry, know how to find the area of a lune (biangle) and small
triangle (using Girard’s Theorem).
7. In Hyperbolic Geometry, know how to represent lines, segments, angles, and
polygons using the Poincaré disk model.
8. In Hyperbolic Geometry, know how angles are measured. On the Poincaré disk
model.