### Test 3 Study Guidelines

1 8.5 x 11 sheet with writing on both sides allowed. You may put anything
you want that fits on your sheet.
I will give you copies of the Appendix in Sibley so
you do not need to write down Euclid's postulates or propositions.
Calculator allowed. You may also wish to bring your child's ball to class.
Review our activities for Euclidean, hyperbolic, and spherical
geometry that relate to the sum of the angles in
a triangle, the Pythagorean theorem, parallelism, and similarity,
but do not worry about all of the
calculation or proof details aside from those mentioned specifically below
(in the bolded sections). Also see this image.

For example, for the sum of the angles in a triangle,
we did the following:

Sketchpad activities for all three geometries
Escher worksheet for hyperbolic geometry
Constructing a 90-90-90 triangle in spherical geometry
Beachball activity worksheet
Paper folding Euclidean demonstration
The Euclidean proof that the sum is always 180 degrees, and, what
goes wrong with this proof in spherical and hyperbolic geometry
**Be sure that you know the following proofs**:

In Euclidean geometry, the sum of the angles in a general triangle
always equals 180 degrees.
In spherical geometry, the sum of the angles in a general triangle is
always greater than 180 degrees.
Bhaskara's proof of the Pythagorean Theorem in Euclidean geometry
(the one with the large square that has 4 right triangles around a smaller
square).
**Be sure that you know the related examples that could be used to
generate a proof of the following:**

In taxicab geometry, the Pythagorean Theorem sometimes but not always
holds.
In hyperbolic geometry, Euclid's 5th postulate sometimes, but not always
holds.
In spherical geometry, if we use "straight" as the definition of a "line"
then SAS is false.
In taxicab geometry, SAS is false.
**Be sure that you know how to do the following:**

Given a work by Escher, use the clues in the picture to specify what
geometry it is (review our worksheet on Circle Limit 4 -- Heaven and Hell by M.C.Escher 1960)
Use the string argument on the sphere that was related to the Pythagorean
Theorem