Do not write up 6.1.
6.2 a. Prove ITT (on three surfaces), b. prove corollary, c.
prove converse (also give counterexamples on the sphere and reconcile this
by indicating for which triangles on the sphere your proof works).
In between we proved: If two isosceles triangles have the same base,
then the segment joining their top angles is also an angle bisector to
each of them and a perpendicular bisector to their base. Feel free
to use this; do not reprove it.
6.3 a. Describe and demonstrate a compass and straightedge
perpendicular bisector construction. Justify it. Analyse it on
all three surfaces. Strengthen your work by using a Euclidean
compass, which closes when you lift it off the paper. b. same
for angle bisectors. Try to see the two as related.
6.4 Consider different Side-Angle-Side configurations on our three
surfaces. How many triangles do each determine? When 1,
why? When 0, why? When 2, why? When more than two,
why? For situations in which there are two or more, can you restrict
to a subset of triangles on that surface so that there is only one?
6.5 Same as 6.4, but for Angle-Side-Angle.
7.1 Derive a formula for the area of a triangle on a sphere.
Justify all steps. (I recommend following the path we did in class.)
7.2 Derive a formula for the area of a triangle on a hyperbolic
plane. Justify all steps. (I recommend following the path we
did in class.)
7.3 Show all work to derive the possible values for angle sum on all
three surfaces. Pay particular attention to justifying the planar
case. Use the approach we did in class.
7.4 Show all visual work to derive the holonomy of triangles on the
sphere, hyperbolic plane, and Euclidean plane. Be careful about
which prior results you are using.
9.1 Same as 6.4, but for Side-Side-Side.
9.2 Same as 6.4, but for Angle-Side-Side. Find as many particular
cases as you can where you limit to only 1 triangle.
9.3 Same as 6.4, but for Side-Angle-Angle.
9.4 Same as 6.4, but for Angle-Angle-Angle. Explain why the
situation is unsalvagable for the Euclidean plane. Justify your
results for the other two surfaces.
8.1 Prove Euclid's exterior angle theorem for all cases on all surfaces for which it is true, and for those which it is not, prove that it is not. (EEAT: any exterior angle of a a triangle is greater than either remote interior angle.)
8.2 Consider two lines that are parallel transports along a third line. Discuss any symmetries of this entire figure of three lines. What can you say about the two parallel transported lines? Do they intersect? If not, why not? If so, where? On all three surfaces.
8.3 [I believe David's questions are well written here, mostly this is just rewriting them.] Prove on all surfaces: if two lines are parallel transports along a third line, then they are also parallel transports along any other transversal (i.e. that crosses both lines) through the midpoint of the segment between the two lines. Are there other parallel transport lines on the sphere and hyperbolic plane? Prove: two lines are parallel transports if and only if they have a common perpendicular. Is the common perpendicular unique?
8.4 First six T/F questions: Parallel transport lines on H^2/S^2 do not intersect; any transversal has congruent corresponding angles for parallel transport lines on H^2/S^2; parallel transport lines on H^2/S^2 are everywhere equidistant. Fully justify your six T/F answers. Then show that there are pairs of lines on H^2 that do not intersect but are not parallel transports. Do NOT write up 8.4c.
10.1 Prove on the Euclidean plane: If two lines are parallel transports, then they are parallel transports along any transversal. Use as few assumptions as possible, and identify your assumptions explicitly. Since this is not true on the other surfaces, why not? This question is better if your assumption is more interesting. Thoughtful creativity is appreciated.
10.2 Logically prove the equivalence of the following: PT!, triangle angle sum=π, H = 0, and parallel transported lines are equidistant. Include your assumption in 10.1 if it was not about intersecting lines. There should be many steps to this to get it all logically equivalent.
10.3 Logically prove the equivalence of the following: Euclid's Fifth Postulate, the high school postulate, and your assumption in 10.1 if it was about intersecting lines. Then logically prove the equivalence of the group in this question to the group in 10.2. In the end you will have a list of six or seven statements which are all logically equivalent. They are different versions of what makes the Euclidean plane special or weird.
10.4 This question returns to H^2 and S^2 and asks what we can say about
parallelism there.
a. Show Euclid's fifth postulate is true on the sphere,
and that if the sum of the interior angles is less than a straight angle
on one side that the lines intersect closer on that side. Be precise
about what you mean by closer.
b. Show that given any point not a line in H^2, there
is a minimal angle of non-intersection, i.e. the smallest angle to the
perpendicular so that at that angle and any greater than it will produce a
non-intersecting line.
c. Use parallel transport to change the high school
postulate to something true on S^2 and H^2. Change as little as you
can, and keep uniqueness.
d. Adapt your 10.1 assumption to H^2 or S^2.
12.1 Both problems on the Euclidean plane: show that every triangle is equivalent by dissection to a parallelogram with the same base (for all bases), and show that every parallelogram is equivalent by dissection to a rectangle with the same base and height (for either choice) [This will require AP].
12.2 [We may switch top and bottom from the book:] Prove that the top angles of a Khayyam Quadrilateral are congruent. Prove that the perpendicular bisector of the base of a Khayyam Quadrilateral is also the perpendicular bisector of the top. Show that the top angles are greater than a right angle on sphere and less than a right angle on a hyperbolic plane. Finally show that a Khayyam Quadrilateral on the Euclidean plane is a rectangle and a Khayyam Parallelogram on the Euclidean plane is a parallelogram.
12.3 This is the analogue of 12.1 after learning 12.2 - so now on all surfaces: show that every triangle is equivalent by dissection to a Khayyam parallelogram with the same base (for all bases), and show that every Khayyam parallelogram is equivalent by dissection to a Khayyam quadrilateral with the same base (use David's definition of base) and height [This will require AP].
13.1 On the Euclidean plane, show that every rectangle is equivalent by dissection to a square.
13.2 On the Euclidean plane, show that the (disjoint) union of two squares is equivalent by dissection to another square.
13.4 Prove: If two triangles have congruent corresponding angles, then the corresponding sides of the triangles are proportional. If two triangles on a plane have an angle in common and if the corresponding sides of angle are in the same proportion to each other, then the triangles have corresponding angles congruent. Finally prove SSS similarity - if two triangles have proportional sides then the corresponding angles are congruent.
Constructions: Complete TWO of 20, 25,
30, and 34. Include computer constructions for each one. Also
include justifications for claims.