### 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).
• 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.
• There will be various types of questions on material up through September 30th related to Geometric Perspectives, Proof Considerations & IGS Exploration and your grade will be based on the quality of your responses in a timed environment (turned in by the end of class).

Topics: Review the following and be sure that you could answer related questions on these topics:
• Eratosthenes from Project 1
• Euclidean proof of I-1 [Construct an equilateral triangle] and what happens on the sphere from class
• Euclidean construction of I-9 [Bisect an angle] from Project 1
• Euclidean proof of I-4 [SAS] and a counterexample on the sphere from class
• Euclidean proof of AAA from Theorem 4.4.5 on p. 149-150 and a counterexample on the sphere from Project 2 and 3
• Counterexample of SSA in Euclidean geometry from class
• Reasons why HL in Euclidean geometry provides congruence from class
• Squares in Euclidean geometry and what happens on a sphere from Project 2
• The Pythagorean theorem in Euclidean geometry (from Euclid's Elements proof and related IGS explorations to the Zhou Bi Suan Jing or Chou Pei Suan Ching proof and puzzle) and what happens on a sphere from Project 2 and class
• Counterexamples of SSSS in a Euclidean quadrilateral from class
• Midpoints of Euclidean quadrilaterals from Project 3
• Models of Rowing a Regatta from Project 3
• Fill in the details and reasons using Euclid's Elements Book I, trigonometry, and the triangle similarity theorems and definitions from class from worksheet on Geometric Modeling Using Similarity

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...
• Does this construction 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.
• Identify applications.
• Give reasons why or why not.
• Describe one of the Interactive Geometry Software explorations.

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
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
Construct a line of a given length starting at a point off the given length and continuing along a line through that point: Prop 3
Bisect an angle: Prop 9
Construct perpendiculars: Prop 11 or 12
Congruence Theorems: Prop 4: SAS, Prop 8: SSS, 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