Comparing Planar and Spherical Geometry

Complete the table below to compare and contrast lines in the system of plane Euclidean geometry andlines (great circles) in spherical geometry.

On the plane On the sphere1. Is the length of a line finite or p- .. .•~infinite? ~- J:

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2. Describe the shortest path

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3. Can you extend a lineforever?

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4. How many parts (and are v

they finite or infinite) will two

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~~ Apoints divide a line? Os

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5. How many lines pass through ~4.~ ~-~:;( 0"1'\..t-any two different points?

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I6. How many lines are parallel

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to a given line and pass through ~~dtY.a given point not on the given ~line?

~ _~. ~~ ~ A#..8A'6,~t- - . 7 v U7. If three points are collinear, ~ .fIr 0 - t~• • oexactly one is between the other ~. (~-'-Dtwo. (True or false)

AFor each property listed from plane Euclidean geometry, write a corresponding statement for sphericalgeometry.

8. Two distinct lines with no point of intersection are parallel.'Two d\st\t\c-\- \irl26 w,~no ~o,V\'\-- o~ ,,,"'~V"sec\-\oV\ do V\c-t e')(\~t

Two distinct intersecting lines intersect in exactly one point.TUlO dis;'\~c.-t \V\"\tV"sec:HV\~H~e~ \~{eysec:-\ ,y) e)(.o..c.\\~ -\-wo f><'i "'~S

I O. A pair of perpendicular lines divides the plane into four infinite regions.~ ?o.\f i)~ 1. \iY\e.s div\ de.$ ""he. spheye ir'\+O 4 -C, W\\-\~ V"e.~ionj

11. A pair of perpendicular lines intersects once and creates four right angles.flr ~oJr (')~ ~ \\Y\e~ l,,-\erse.c\-s +wice. ~",d. cre.cd·e5 & r,'&h+ ~\es

12. Parallel lines have infinitely many common perpendicular lines.Ya.Y'o.\\~\ \'nes do n()~ e".'ls+.

13. There is only one distance that can be measured between two points.'\ ""er e. 0..(' e. 1-w0 c!is-\ Q.\'\c e s ¥no..-\ C0.\1\ be YrI e Q.SU r Q d

14. There is exactly one line passing through two points. w. p~'n'''\-''S,heft is e ~o..c\-\~oW\e. \t Y\e. 'Po..SS\ Y\~

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9.

.• -.Choose one of the fol owing answers for each question: A) true on a plane

B) true on a sphereC) true on both a plane and a sphereD) true on neither a plane or a sphere

e 15. Aline is an infinite set of points. ~? ~

c... 16. A line is continuous (no "holes" or gaps).

A 17. Through any two points, there is exactly one line.

..B 18. There exists at least one pair of points through which more than one line can be drawn.

~19. A polygon may have two sides.

A 20. Each angle of an equilateral triangle must be 60· .

D 21, Each angle of an equilateral triangle may be 450

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~ 22. Each angle of an equilateral triangle may be 1200

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B

.B 23. A line is bounded. (that is, it can fit into a closed box)

A 24. There is no greatest distance between two points.

A 25. Two lines can share no points.

B 26. Two distinct lines can share two points.

D 27. Two distinct lines can share more than two points.

~28. The sum of the angles ofa triangle is always the same number.

A 29. A triangle can have at most one right angle.

-B-30. Three lines may be perpendicular to each other (that is, line a ..Lline b .lline c .lline a)

A 31. Three lines may be parallel to each other.

~32. Three lines may intersect in three points. (each of the lines intersects the other two lines)

~33. Three lines may intersect in two points. (each of the lines intersects the other two lines)

~34. Three lines may intersect in four points.

~35. Vertical angles are congruent.