Assignment 7 (Due Wed 3/3)

Read Sved Chapter 4 and Learn about the P-model in Chapter 5.

7.1 Constructions in the P-model.

Construct the following in the Poincare model with GSP or by hand and turn in.

Also, be prepared to explain or carry out any of these constructions in class on Wed.

All points are P-points and all lines are P-lines unless otherwise specified.

In the P-model: Construct these figures. (Nice quality, clear indication of how they work.)

1. Construct the line through 2 points A and B.

2. For a point C not on line AB, construct the two lines through C that are converging parallel to line AB.

3. For a point C not on line AB, construct the line through C perpendicular to line AB.

4. For a point C not on line AB, construct the reflection of C in line AB.

5. For two (random) points A and B, construct the perpendicular bisector of AB.

6. Construct two ultraparallel lines and then construct the common perpendicular to these lines.

7.2 Angular Defect is Additive

• Show that the angular defect is additive. This means that if a triangle is partitioned into two triangles, then its angular defect is the sum of the angular defects of the sub-triangles.
• Does this proof also work on the Euclidean plane and the sphere? What does this say in each case?
• Explain why, if one rectangle exists in a plane, then arbitrarily large rectangels exist.
• Explain why, if one triangle with positive angle defect exists in a plane, then no rectangles can exist.

Read Sved Chapter 4 to learn about angular defect.