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
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.
The construction of Proposition 1 [Construct an equilateral triangle] in
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
[summarize how the development and how the example/topic illustrated this]
and [a separate question] connections among multiple mathematical
[which perspectives, and how the example/topic illustrated these].
Review the following and be sure that you could answer related questions
on these topics:
and spherical geometry from class
construction of Proposition 9 [Bisect an angle] from Project 1
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.
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].
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