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