### Exam 2

### At the Exam

You may have your child's ball with you.
You may have out food, hydration, ear plugs, or similar if they will
help you (however any ear plugs must be standalone--no cell phone, internet or other technological connections).
Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.
This exam is closed to notes/books and closed to technology, but I will give you a copy of *Euclid's Elements* Book I.
Any IGS explorations I ask you to describe or create will specify "roughly sketch" so a sketch by-hand
without tools will be fine, however, if you prefer, you may bring a straight edge and/or a compass or circle.
Your grade will be based on the quality of your responses in the timed environment.
Because there is more material on this exam than exam 1,
you may make yourself some reference notes on the small card I hand out (additional cards are on my door if you need to
rewrite it). The reference card must be handwritten. Think of the card as a way to include some important
content that you aren't as comfortable with. You won't have room for everything, and you should try to
internalize as much as you can.

**Topics**:
Review the following and be sure that you could answer related questions
on these topics.
The vast majority of the exam will come from a variety of types of questions related to some of these topics. However, exams are not only an opportunity for you to demonstrate your mastery of the material, but are also an opportunity for you to be challenged with new material in order for you to make new connections.

**Specific Examples of Types of Questions**
Question types include short answer/short essay, like:
Sketch the construction or diagram...
Does this construction or diagram always, never, or sometimes (but not always) work on the sphere? Explain.
Sketch or provide counterexamples...
In the following proof, fill in the blank using reasons from Book 1 of *Euclid's Elements* (which I will hand out to you) and identify any additional underlying assumptions.
Write a paragraph proof and identify underlying assumptions/limitations.
What goes wrong with the proof in hyperbolic geometry?
Identify applications.
Give reasons why or why not.
Describe one of the Interactive Geometry Software explorations.
How did we explore this in class?

*Euclid's Element's* Book I:
I will give you a copy 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 [fails in spherical geometry and
taxicab geometry since uniqueness of lines was used in Prop 4,
even though it is not explicitly stated here]

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 [fails in hyperbolic
geometry]

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

Exterior angle is larger than each remove angle: Prop 16

Recognizing a parallel: Prop 27

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