Name: _______________________________ Date: ______________ Class: ____________

Postulates and Theorems to be Examined in Spherical Geometry

Some basic definitions:1. Line segment: The segment AB, AB , consists of the points A and B and all thepoints on line AB that are between A and B.2. Circle: The set of all points, P, in a plane that are a fixed distance from afixed point, O, on that plane, called the center of the circle.3. Parallel lines: Two lines, l and m on the plane are parallel if they do not intersect.4. Sphere: The locus of the points in space that are a given distance from a fixedpoint, called the center of the sphere.5. Great circle: A great circle is a circle whose center is the center of the sphere andwhose radius is equal to the radius of the sphere.6. Arc of a great circle: The shortest path between two points on the sphere is the arc of a

great circle.7. Geodesic: Lines in geometries other than the Euclidean plane. On the spherethis is a great circle. On the Poincaré model of the hyperbolic plane itis the inscribed arc of a circle orthogonal to the boundary of thePoincaré circle.8. Antipodal points: Points that lie at the intersection of a great circle and a line throughthe center of the circle on the sphere.9. Small Triangle: The small triangle is formed by joining three non-collinear verticeswith the shorter arc between the vertices. Three vertices thendetermine only one spherical triangle.The following postulates will be examined:1. There exists a unique line through any two points.2. If A, B, and C are three distinct points lying on the same line, then one and only one of thepoints is between the other two.3. If two lines intersect then their intersection is exactly one point.4. A line can be extended infinitely.5. A circle can be drawn with any center and any radius.6. The Parallel Postulate: If there is a line and a point not on the line, then there is exactly oneline through the point parallel to the given line.

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7. The Perpendicular Postulate: If there is a line and a point not on the line, then there isexactly one line through the point perpendicular to the given line.8. SAS Congruence Postulate: If two sides and the included angle of one triangle are congruentrespectively to two sides and the included angle of another triangle, then the two trianglesare congruent.The following theorems will be explored:1. Vertical Angles Theorem: Vertical angles are congruent.2. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.3. Theorem: If two lines are parallel to the same line, then they are parallel to each other.4. Theorem: If two lines are perpendicular to the same line, then they are parallel.5. Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180o.6. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sumof the measures of the two nonadjacent interior angles.7. Third Angles Theorem: If two angles of one triangle are congruent to two angles of anothertriangle, then the third angles must also be congruent.8. Angle-Angle Similarity Theorem: If two triangles have their corresponding anglescongruent, then their corresponding sides are in proportion and they are similar.9. Side-Side-Side (SSS) Congruence Theorem: If three sides of one triangle are congruent tothree sides of a second triangle, then the two triangles are congruent.10. Angle-Side-Angle (ASA) Congruence Theorem: If two angles and the included side of onetriangle are congruent to two angles and the included side of a second triangle, then the twotriangles are congruent.11. Theorem of Pythagoras: In a right triangle, the square on the hypotenuse is equal to the sumof the squares of the legs.

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Points, Lines, and Triangles in Spherical Geometry

1. Spherical geometry is geometry on a sphere. What is a sphere?

rr

2. Locate two points in the plane and label them P and Q.a. What is the shortest path between these two points?b. Stretch a piece of ribbon between the two points to indicate the shortest path.

P Q

3. Locate two points A and B on the sphere. Choose points that are NOT opposite each other onthe sphere (antipodal). Use a piece of ribbon to find the shortest path between the twopoints on the sphere just as you did on the plane. Describe what this path looks like.A B

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4. Extend segment PQ in the plane. How far can you go?P Q

5. Extend the segment on your sphere. How far can you go?6. You have just drawn a great circle.a. Define a great circle.

b. Name a great circle on Earth.c. Are all lines of latitude great circles? Explain your answer.

7. In the Euclidean plane the shortest path from P to Q is unique, and its measure is fixed. Canthe same be said of the segment AB on the sphere? Is the measure of AB unique? Explainyour answer.

A B

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8. Measure the distance between two points P and Q in the Euclidean plane.a. What instrument did you use to measure the distance?b. What units did you measure distance in?c. Could you use the same instrument to measure the distance between A and B on thesphere?

9. The Earth as a sphere in Euclidean space has a radius of 6,400 km.a. What is Earth’s circumference?b. How many degrees does this represent?c. If two places on Earth are opposite each other, what is the distance between them inkilometers in the spherical sense? In degrees?d. If two places are 90° apart from each other, how far apart are they in kilometers in thespherical sense?e. If two places are 5026 km apart, what is their distance apart measured in degrees?f. Mars has a circumference of 21,321 kilometers. What does this distance represent indegrees?g. What is the furthest distance that two places on Mars can be apart from each other indegrees? In kilometers (in the spherical sense)?

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Construct a simple spherical ruler for the sphere as follows:

Cut strips of paper (the thicker the paper the better since the straight edge of thepaper will be used as a ruler) and lay them side by side so that the strip formed is longenough to form a great circle. The width of the strips will depend on the size of thesphere you are using. It is important that the strips be cut thin enough so that whenthe resulting circle is placed on the sphere it lies flat and does not have wrinkles in it.The strips can be attached side by side with pieces of tape. Once the strips are joined in such a way that they fit snugly around the sphere, we usea marker to divide the circle into quarters, and label 0o, 90o, 180o, and 270o. Bycontinuing to fold the circle carefully, it is possible to mark off intervals of 10o on thestrip.

10. Euclid’s first postulate states that for every point P and every point Q, where P is not equalto Q, there exists a unique line l through P and Q. Is this postulate valid in sphericalgeometry? Justify your answer.11. Draw two lines on the plane. In how many points do these two lines intersect?

12. Draw two great circles on the sphere. In how many points do two lines on the sphereintersect?

A

B

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13. Euclid’s parallel postulate states that if l is a line and P is a point not on l, then exactly oneline can be drawn through P that is parallel to l. Can you re-word this postulate so that it istrue for spherical geometry? Explain.

14. Locate a point R between two points P and Q on the plane.P QR

The Betweenness Axiom states that if P, Q, and R are three points in the plane, then one andonly one point is between the other two.Locate a point C between points A and B on the sphere. Is the Betweenness Axiom valid forthe three points that are drawn on the sphere? Justify your answer.

BA C

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15. Euclid’s second postulate states that a line segment can be extended infinitely from eachside. Is this postulate valid in spherical geometry? Justify your answer.

16. Draw a circle on the plane. Locate the center of the circle. Define the term circle. Measurethe radius of the circle.r

17. Draw a circle on the sphere. Locate the center of the circle.

a. How many centers are there?b. Draw in the radius of the circle. Compare the radius of the spherical circle with thatof the planar circle.c. Measure the radii of the spherical circle. How is one related to the other?

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18. Euclid’s third postulate states that a circle can be drawn with any center and any radius. Isthis true for circles on the sphere?

19. Draw two intersecting lines on the plane. Use a protractor to measure the vertical angles.Confirm that the vertical angles theorem is valid on the plane.P

QR

S

O

20. Draw two great circles. Label the points of intersection A and B. Use your sphericalprotractor to measure the pairs of vertical formed at the point of intersection of the greatcircles. What do you notice?

A

B

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21. Draw a Euclidean line.a. Locate a point P that is not on the line. What is the shortest path from the point tothe line?b. This path is called the distance from P to the line. Construct this path. How manysuch paths can you construct?

R Q

P

S

22. Draw an arc on the sphere.a. Locate a point P that is not on the arc. What is the distance (shortest path) from thepoint to the arc?b. How many perpendiculars can be drawn from the point P to the arc?c. Complete the great circle associated with this arc and locate a pole point for thisgreat circle. How many perpendiculars can be drawn from this pole to the greatcircle?

P

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23. Draw two intersecting lines l and m on the plane. Can you draw a common perpendicular tothese two lines?24. Draw two parallel lines l and m on the plane. Can you draw a common perpendicular tothese two lines? If you can, how many can be drawn?

25. Draw two great circles that are not perpendicular. Can a perpendicular be drawn to thesetwo great circles?

l

m

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26. In how many different ways can three lines intersect in the plane?

27. In how many points do two geodesics intersect? Explain your answer.

28. In how many ways do three geodesics on the sphere intersect?

m

k

lm

k

l

l

m

k

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29. What is the minimum number of sides required to draw a closed figure in the plane usingstraight lines only? Name the figure you drew in the plane.

Q

P

R30. What is the minimum number of sides required to draw a closed figure on the sphere?What do you think we should call the figure on the sphere?

a. You may have decided that the term biangle would be appropriate for this shape. Howmany biangles are formed by the intersection of two great circles?b. Another name for this figure is a lune. What is the relationship between the two pointsof intersection of the sides of the lune?c. How long are the sides of the biangle?d. Measure the opposite angles of the biangle. What do you notice?

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31. Construct three non-collinear points in the plane. Connect them to form a triangle. Howmany triangles can you form?

32. Locate three non-collinear points A, B, and C, on the sphere.

a. Use different colors to join two points at a time. How many different arcs join thepoints together?b. How many different triangles with vertices A, B, and C, can be drawn? (Use ofdifferent colors may help to identify the triangles more easily.)

We will define the triangle formed using the shorter arcs joining two points on the sphere as thesmall triangle. Identify and shade in the small triangle on the sphere.

A

CB

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33. Draw a triangle on the plane. What is the sum of the interior angles of the triangle?

34. Locate three points A, B, and C on the sphere so they form a small triangle.

a. Measure the angles of the triangle.b. What is the sum of the measures of the angles of the triangle?c. Draw another larger triangle and measure its angles and find the angle sum.

d. Is the angle sum the same for both triangles?

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35. Is it possible for a triangle on the plane to have more than one right angle?

36. Is it possible for a triangle on the sphere to have more than one right angle?

37. Draw a triangle PQR on the plane. Extend side QR to S. PRS is an exterior angle of thetriangle. What is the relationship between PRS, P, and Q ?P

Q R S

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38. Draw ABC on the sphere. Extend BC to D and measure ACD . Measure A and B . Is there a relationship between the exterior angle of a triangle on the sphereand the non-adjacent interior angles?

A

B DC

39. Draw a ΔPQR in the plane. Measure the size of each angle of the triangle. Construct ΔXYZwith P X, and Q Y and 2XY PQ and 2YZ QR .a. How does the measure of the third angles of the triangles compare?b. How do the measures of PR and XZ compare?

Q

P

RY

X

Z

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40. Draw ΔABC on the sphere. Measure the size of each angle of the triangle. Construct a secondΔDEF with A D and B E and 2DE AB and 2FE BC . Measure the thirdangle of the triangle and compare this measure with the measure of the third angle oftriangle ABC.

41. In the plane, the Third Angles Theorem states that if two angles of one triangle arecongruent to two of another, then the third angles are congruent. Does this theorem applyto triangles on the sphere?

42. State the theorem of Pythagoras with respect to triangle PQR in the plane.

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43. Investigate whether this theorem is relevant on the sphere. We have already discoveredthat 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 anyrelationship between the sides of the triangle.

44. What is the definition of similar polygons?

45. Suppose we construct PQR on the plane. We measure its angles and the lengths of thesides. Then we construct a second XYZ , with sides double the length of ΔPQR, and with m P = m X and m Q = m Y . We measure the sides of PQR and find the ratios, , andPQ QR PR

XY YZ XZ. What do you notice about the values of the ratios?

P

Q R Y Z

Xm QR = 1.34 inches

m PQ = 1.17 inches

m RP = 1.24 inches

m PQR = 58°

m PRQ = 54°

m YZ = 2.68 inches

m XYZ = 58°

m XY = 2.34 inches

m XZ = 2.45 inches

m QPR = 68°

m YXZ = 68°

m YZX = 54°

m PQm XY

= 0.50 m QRm YZ

= 0.50 m RPm XZ

= 0.51

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46. Suppose we draw a triangle on the sphere and measure the angles of the triangle and thelengths of the sides. Then we construct a second triangle with angles equal in size to theangles of the first triangle, and we measure the sides of the triangle. What do you notice?

47. Construct a triangle on the plane using convenient lengths for the sides of the triangle.Construct a second triangle with sides identical in length to the sides of the first triangle.Are the two triangles congruent?

48. Construct a triangle on the sphere. Measure the length of the sides of the triangle. Constructa second triangle with sides equal in measure to the sides of the first triangle. Measure theangles of the two triangles. Are the two triangles congruent?

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49. Draw ΔPQR on the plane. Measure the length of side PQ and QR and the degree measure ofPQR . Repeat this construction for a second ΔXYZ with the measure of PQ = XY, QR = YZ,and PQR XYZ . Are the two triangles congruent?

50. Draw ΔABC on the sphere. Measure the lengths of the sides AB and BC and the measureof ABC . Repeat this construction for ΔDEF where the measure of AB = DE, BC = EF andABC DEF . Are the two triangles congruent?

51. Construct ΔPQR on the plane. Measure the length of QR and the measure of Q and R .Construct ΔXYZ with the measure of QR = YZ and Q Y and R Z . Are the twotriangles congruent?

52. Construct ΔABC on the sphere. Measure the length of BC and the measure of B and C .Construct ΔDEF with the measure of BC = EF and B E and C F . Are the twotriangles congruent?

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53. Write down the formula for the surface area of a sphere.54. Construct a biangle with angle of 600. What is the area of this biangle?

55. Construct a biangle with an angle of 900. What is the area of this biangle?

56. Write down a generalized formula for the area of a biangle.60 90

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57. Write down the formula for the area of a triangle in the Euclidean plane.58. If two triangles have identical angles, are they necessarily congruent?59. We will now derive a formula for the area of a triangle on the sphere. In order tounderstand how the formula is derived, you are encouraged to draw triangles on the sphereand use colors to identify the different triangles under discussion. This is very important toa clear understanding of the derivation of the formula for the area of a triangle on thesphere. This formula is commonly known as Girard’s Theorem.

Draw a triangle on the sphere and label the angles , , and as shown in th diagram below.a

b

g

A

B

C

Using colors , draw the α-lune. Notice that there is a congruent α-lune on the back of the sphere.Repeat this for the β-lune and the γ-lune. 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. If wenow wished to get an expression for the area of the sphere in terms of the area of the lunes, wewould get the following (luneα = area of lune α)

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Area of sphere = 2 luneα + 2luneβ + 2luneγ - 4ΔABC2 2 2 2

2 2 2 2

2 2 2 2

2

4 2( )4 2( )4 2( )4 4360 360 360

4 2( )4 2( )4 2( )4 4360 360 360

( ) ( ) ( )180 180 180

180( )180

r r r r ABC

ABC r r r r

ABC r r r r

ABC r

This formula has a very interesting consequence for the area of a triangle on the sphere. It statesthat the area of a triangle on the sphere is directly related to the angles of the triangle.The formula

2 180( )180

ABC r is called Girard’s Theorem, and the quantity

180o is called the spherical excess of the triangle.

60. Calculate the area of a spherical triangle with angles of 90o, 90o, and 90p.

61. Draw these triangles on the sphere and confirm that the answer you got for the area isconsistent with what you would have expected starting with the formula for the sphere,24A r

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62. Investigate the area of a polygon using the diagram below, and determine whether aformula for calculating the area of an n-sided polygon exists.

A

B

C

E1

2

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