- In Problem 4: we were directed that we could use
the first 28 propositions of
Euclid and Theorem 1.2.1 but nothing else, to prove the equivalency of:

*Statement a)*: If a straight line cuts one of two parallel
lines, it cuts the other.

*Playfair's*: There is exactly one parallel to a line through a
given point

Which of the following would satisfy Sibley's assignment?

a) For Playfair's ---> statement a, assume a straight line cuts
one of two parallel lines. If, for contradiction, it doesn't cut the
second parallel, then we would have 2 parallels through the intersection,
contradicting Playfair's, so statement a holds.

b) For statement a ---> Playfair's, create the parallel that is the
perpendicular to the perpendicular (using I-12, I-11 and I-16), and then
any other line that cuts the parallel won't make the same side interior
angles equal to 180, so it can't be parallel, showing Playfair's.

c) Both a) and b)

d) Neither a) nor b)

- Which do you find most compelling about why Euclid wrote the 5th
postulate the way he did?

a) Euclid's 5th is more of a self-evident truth than Playfair's

b) In Prop 31, Euclid constructs a parallel, but he doesn't use
the language of uniqueness (there can be only one) in Book 1

c) Euclid was trying to keep the same kind of language as the other
postulates

d) It is easier to use Euclid's 5th in the propositions to help
prove and support them than it would be in using Playfair's

e) There was no notion of infinity then, so instead of Playfair's
which refers to never intersecting, Euclid's 5th gives something
constructive about intersection.

- Which of the following are true from Problems 2 and 3 in solutions

a) In Euclidean geometry, the exerior angle is exactly equal to the remote interior angles, in hyperbolic geometry the exterior angle is too big, because the triangle sides bow in, creating a smaller angle sum, and on the sphere the exterior angle is too small, because the triangle sides bow out, creating a bigger angle sum

b) Proclus assumed that the lines were a constant distance. This
is a problem, because
in hyperbolic geometry, there are examples where a transversal to one
parallel never overtakes the distance between the parallels.

c) Both a) and b)

d) Neither a) nor b)
v

- Would you like me to have an ASULearn chat session for Monday at 8pm?

a) Yes and I can make it then

b) Yes but I might not attend (or will not attend)

c) Unsure

d) It is unnecessary