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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 4ABC 360 360 360 Switch triangle and sphere formulas: 4ABC 2( )4 r 2 2( )4 r 2 2( )4 r 2 4 r 2 360 360 360 4ABC ( )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 ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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 ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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. _______________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ Is the measure of a segment AB unique on a sphere? Explain your answer. ____________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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? _____________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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. ___________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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°.________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 30. Determine a formula for calculating the area of an n-sided polygon. __________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 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.