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## Learning Goals for Today

Students will be able to:

1. State the Parallel Postulate in at least two forms (e.g., Euclid and Playfair) and explain its meaning (uniqueness of parallels).
2. Provide a written summary of the role of the Parallel Postulate in development of Euclidean geometry.
3. Explain in writing how the theory of parallelograms fits into the development of Euclid Book I.
4. Present (at the board or in writing) two different rigorous explanations of the traingle area formula A= bh/2 and relate it to the formula for the area of a parallelogram.
5. State and prove Euclid I:35.

## Agenda

1. Quiz (5 min.)
2. Discussion of test (February 22) (5 min.)
3. Lecture/discussion on the Parallel Postulate (25 miniutes)
• What is says (uniqueness) and what it does not say (existence).
• What is "Neutral Geometry"? What is "Non-Euclidean Geometry"?
• Consequences of the Parallel Postulate.
• The role of the Parallel Postulate in Euclid Book I: Parallelograms and the theory of area.
4. Triangle Area (25 minutes)
5. Homework discussion (20 minutes).

## Post-class report on what we did

We covered items 1 and 3 in the agenda fully. Item 2 was not taken up because students had not prepared question proposals. (Maybe this needs to be explicit in homework assignment.) Items 4 and 5 were treated only briefly. The goals in red were fully addressed. The goals in violet were only partially addressed.

During class, proofs of the following theorems were presented:

1. Two lines perpendicular to a third are parallel to one another. The proof does not use the Parallel Postulate. It does use Euclid I:17.
2. Opposite sides of a parallelogram are congruent. The proof DOES use the Parallel Postulate (and ASA).

Also, Stillwell 1.3.6 was discussed during the last 10 minutes of class.

## Homework

1. The due date for problems originally to be handed in 02/08/06 has been extended. The new due date will be Friday, February 10. Instructions for turning in the assignment will be given in class on Wednesday. Class on Wednesday will be devoted to assisting you with the solutions to these problems. The assignment is:
• Stillwell 1.3.6, 1.4.1, 1.4.2, 1.4.3, 1.4.4, and the problems on the Sea Island Page here.
2. By Monday February 13, read and study the readings at the Traingle Area Page. Browse the pre/post tests from a previous class, and determine which methods of proof or explanation were used in the post-tests.
3. Due Monday February 13: bring to class two questions that you would consider appropriate for the first test.
4. Due Wednesday February 15: Stillwell 1.5.1--4 (page 17).