The following is NOT HOMEWORK unless you miss part or all of the class. See the Main Class Web Page for ALL homework and due dates.

Thanksgiving Break

1. What is the sum of the angles in a hyperbolic triangle? How large can the sum of the angles get? How small can the sum of the angles get?

2. The sum of the angles in a spherical triangle

3. Theorems from Perry p. 247-248 and p. 255 in hyperbolic geometry - Do these in Sketchpad.

Hyperbolic Parallel Axiom: If m is a line and A is a point not on m, then there exist exactly two noncollinear halflines AB and AC which do not intersect m and such that a third halfline AD intersects m if and only if AD is between AB and AC.

Theorem 5: If AB is a parallel halfline for a line m from point A and AK is the opposite halfline for AB, then AK is not parallel to m.

Theorem 6: If AB is a parallel halfline for a line m from point A and AK is the opposite halfline for AB, then AK does not intersect m.

4. Is the Pythagorean theorem ever, always or never true in hyperbolic geometry?

Review material together as a class and if time remains, then work on final projects.

Image of Sum of Angles

Sketchpad of Sum of Angles

Image of Hyperbolic Pythagorean Thm

Sketchpad of Hyperbolic Pythagorean Thm

Part 1: Choose roles so that one person will hand measurements to the other: Try to give your partner 2 or 3 measurements and instructions that will always ensure that they will be able to draw a triangle with the same shape as yours. Note: Since we only require similar and not congruent triangles, pairs of corresponding sides you give to your partner must only be proportional (via the same proportionality constant) to your measurements, while angles should be equal. Try to come up with a complete list of all of the possible sets of 2 or 3 measurements and instructions that will always result in similar triangles.

Part 2: Switch roles so that the person who received measurements in part 1 will now hand measurements to the other: Try to give your partner 2 or 3 measurements and instructions so that they will not necessarily be able to draw a triangle with the same shape as yours. The person who is given the measurements should try and draw 2 differently shaped triangles from the measurements and instructions. Try to come up with a complete list of all of the possible sets of 2 or 3 measurements and instructions that will NOT ensure that they have the same shape.

Report back to the class when we come back together.

If finished early, then work on the homework for Wed (see main web page) and/or p. 216 number 4 from Project 3.

Similar Triangles activity from Exploring Geometry with Sketchpad.

Create a segment with the ruler tool.

Using the arrowhead tool, choose one of the endpoints and the segment too (by holding down the shift key as you select them)

Under Construct, use the Sketchpad feature to construct a perpendicular line through the endpoint.

Use the point tool to choose a new point on the perpendicular.

Use the ruler tool to construct the segment between the 2 points on the perpendicular line (ie before you do this, the entire line has been created, but the segment does not exist).

Use the arrowhead tool to select only the perpendicular line (but not the segment you just constructed)

Under Display, release on Hide Perpendicular Line.

Use the ruler tool to complete the third side of your right triangle.

Measure the right angle to verify that it is 90 degrees.

Measure the length of the three sides of the triangle.

Once you have all three lengths, under Calculate, click on the measurement of the base of the triangle in order to insert it into your calculation.

Continue in order to calculate the base*base + height * height - hypotenuse *hypotenuse

Move the points of your triangle around in order to try and verify (empirically) the Pythagorean Theorem.

Under File, release on open and open up MacHD/Applications/Sketchpad/Samples/Sketches/Geometry/Pythagoras.gsp

Go through Behold Pythagoras!, Puzzled Pythagoras, and then Shear Pythagoras. Click on Contents to get to the other Sketches.