Test 1 on Axioms, Constructions, Measurement, and Similarity

This test will be closed to notes/books, but a calculator will be allowed (but no cell phone nor other calculators bundled in combination with additional technologies) and you may also bring your ball. There will be various types of questions on the test and your grade will be based on the quality of your responses in a timed environment (turned in by the end of class). You may also bring a straight edge and compass or circle, but any construction sketches I ask you to create will specify "roughly sketch" so a sketch by-hand without tools will be fine too.

Big Picture Questions: Briefly skim the information on the class highlights page and your notes from the first day of class up through and including activities on similarity and project 2 solutions and think about the "big picture". You will be asked to choose and discuss in detail an example of a topic from class that illustrated concept development [summarize how the development and how the example/topic illustrated this] and [a separate question] connections among multiple mathematical perspectives [which perspectives, and how the example/topic illustrated these].

Content Questions: Review the following and be sure that you could answer related questions on these topics:

  • The construction of Proposition 1 [Construct an equilateral triangle] in Euclidean and spherical geometry from class
  • The Euclidean construction of Proposition 9 [Bisect an angle] from Project 1
  • Bhaskara's Euclidean Proof of the Pythagorean theorem from Project 1 and class work
  • Sum of the angles in a Euclidean and spherical triangle from Project 2
  • The Pythagorean theorem on a sphere from Project 2 and class work
  • Squares on a sphere from Project 2
  • The Euclidean proof of SAS from Euclid and a counterexample on the sphere
  • The Euclidean proof of AAA from Theorem 4.4.5 on p. 149-150 of Wallace and West Roads to Geometry and a counterexample on the sphere
  • SSA in Euclidean geometry
  • Be familiar with the language and organization of the appendix of Sibley The Geometric Viewpoint p. 287-292.

    Specific Examples of Types of Questions Question types include short answer/short essay, like:
  • Sketch the construction...
  • Does this construction always, never, or sometimes (but not always) work on the sphere? Explain.
  • Sketch counterexamples
  • In the following proof, fill in the blank using reasons from Book 1 of Euclid (which I will hand out to you)
  • Discuss in detail an example or topic from class that illustrated connections among multiple mathematical perspectives [which perspectives, and how the example/topic illustrated the connections].

    Sibley's Appendix I will give you a copy of this appendix to use on the test. I may give you a proof and ask you to fill in the reasons with the Postulates and/or Propositions. For example, you should be familiar with the statements of the five postulates, and roughly know where some of the propositions are located, as follows:

    Create a line segment: Postulate 1
    Extend a line: Postulate 2
    Create a circle: Postulate 3
    All right angles are equal: Postulate 4
    How to tell that two lines intersect: Postulate 5

    Construct an Equilateral triangle: Prop 1
    Bisect an angle: Prop 9
    Construct perpendiculars: Prop 11 or 12
    Congruence Theorems: Prop 4: SAS, Prop 26: ASA and AAS
    Statements that use the parallel postulate begin with Prop 29, so if you have "if parallel then ..." generally you will want to look at 29 and beyond.
    If parallel then alternate interior angles...: Prop 29
    Construct parallels: Prop 31
    Sum of the angles in a triangle is 180 degrees: Prop 32
    Pythagorean Theorem: Prop 47 and 48