Study Guide - Exam 2

The exam will be on Wednesday, June 2 and cover the material starting with progress report 5.4 and ending with the Ruler Postulate. The format will be similar to the previous exam: some `circle the best answers', support the statement, prove some statements you should be familiar with, and perhaps a couple of new things to consider. You will have the full lab period to complete the exam. As you can tell from the list, we have covered quite a variety of material! I have listed some of the highlights that you should be familiar with.

From the axiomatic approach to transformational geometry:

• Show that any motion of the form is a quasi-rotation. Be able to use the Join Theorem to establish this result. (This, along with other material related to the final report, will be posted Wednesday afternoon.)

• Be able to show that if the point is not on the line then is a glide reflection.

• Given a sketch with the lines and be able to find the approximate location of the point and the line such that motion

• Use an induction argument to show that the composition of finitely many reflections can be reduced to three or less.

• Be able to use the Euclidean Parallel Postulate in tandem with the axioms of transformational geometry to show that - in the Euclidean plane - if two lines are parallel, then they admit a common perpendicular.

• Be able to answer questions similar to the ones that appeared on the second homework assignment (putt-putt golf and symmetries of a figure.)

• Prove that every motion in an elliptic plane is a rotation.

• Be able to describe the fixed points and fixed lines of any motion in the Euclidean plane. (This is great material for 'circle the best answer' questions.)
  
  
  

Motions in the Euclidean and plane and related material:

• Be able to prove the Pythagorean theorem using similar triangles.

• Know the definitions of sine and cosine and be able to prove the law of sines.

• Be able state the 'vector' definitions of a reflection with axis and a translation.

• Be able to show that if and are parallel lines, then there is a vector such that

• Know the matrix representations of a rotation (of degrees) centered at the origin and the reflection when is a line passing through the origin that makes an angle of degrees with the origin.

• Be able to derive the matrix representation of a rotation of degrees when the rotation is centered at the origin.

• Be able to prove the Ruler Placement Theorem. Know the definitions of function, one-to-one, and onto.

• Given the equation of a line and one of the distance functions or be able to produce a coordinate system for (i.e., a function from to that satisfies the Ruler Postulate.)