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.

Why Study Hyperbolic Geometry? Look at Interactive Euclid's Elements and go through the links for the 5 postulates.

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.

Part 3: Similar Triangles - AA Similarity activity sheet from Exploring Geometry with Sketchpad. Leave the Explore More part until later.

Part 4: Use the Triangle_Similarity.gsp file (control click and save the file. Then open it from Sketchpad) to complete the Similar Triangles - SSS, SAS, SSA worksheet. Leave the Explore More part until later.

Part 5: Then complete the Similar Polygons Sketchpad activity sheet.

Part 6: Go back to the Explore More parts of the worksheets.

Part 7: If time remains then look at the main web page for upcoming homework and work on the models for Tuesday and/or read over Project 5.

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.

Sketchpad has some built in explorations. Under File, release on open and click on Desktop, then on ASU-FS4.Apps, then on Mac Software, then on Math Dept, and then on Sketchpad, then on Samples, then on Sketches, then on Geometry and finally, open Pythagoras.gsp For future reference, I will write this as Desktop/ASU-FS4.Apps/Mac Software/Math Dept/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.

Read through Euclid's Proof http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html along with the appendix of Sibley to try and understand it.

We come back together and go through Euclid's Proof of the Pythagorean Theorem. Discuss the benefits and difficulties of using the different methods, including original historical sources.