Q1: Which Of The Following Would Be Considered A "Line" By Euclid's Definition

MTH 250 Graded Assignment 7, part two

Experiment with hyperbolic geometry

Plotting and measuring: Non-Euclid applet at

http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html

You can set the background to white under View>>Background. This may work better for copying/printing.

Note: Applet has printing problems, and copy/paste is not available. Grr. The trick to doing this is using the "Print Screen" key- usually in the upper right of your keyboard. "Print Screen" (or "PrtScr" or similar) copies what is on your screen to the clipboard, then you can paste as usual into Word. Holding down Alt+PrtScr generally will copy just the active window (try it when you're on the Non-Euclid window).

Q1: Start off by drawing and measuring a few triangles.

 Select Constructions >> Plot Point, and plot 3 points by clicking.  Select Constructions >> Draw Segment, and click endpoints to connect (A to B, B to C, C to A).  Select Measurements >> Measure Triangle, and it will give all three side lengths and angle measures.  Select Edit >> Move Point, drag the vertices of your triangle around, and watch the measurements change. The angle sum appears to always be less than ???

Q2: Draw a pair of parallel lines by constructing alternate interior angles equal. You can select File >> New to clear the previous construction

 Constructions >> Draw Infinite Line. Click on two spots on the disk, and you'll get points and a line.

 Constructions>>Draw Ray at Specific Angle.  Enter a value for the angle (default is 45)  Click B, then A, then somewhere on the disk.  You'll get ray AC and ray AB forming an angle.  Constructions>>Draw Ray at Specific Angle.  Keep the same value as above.  Click A, then C, then somewhere on the disk, to the side of ray AC which is opposite B. (You want alternate angles).  You should get a ray CD.  You may need to extend ray CD into an infinite line by choosing Constructions >> Draw Infinite Line, and clicking points C and D. * print or print screen / paste this image*  The lines should be non-intersecting; i.e. parallel.  Measurements >> Measure Angle and measure angles BAC and DCA to verify that alternate interior angles are equal.  Now, Edit >> Move point  Slide point C around. The angles should stay fixed by construction. Can you ever get line CD to intersect line AB, or are they always parallel? (Being the same line doesn't count!)

Q3: A parallel to a parallel may not be parallel!

 Draw infinite line AB.  Constructions >> Draw Perpendicular to line AB through point B by first clicking point B, then the line.  Same thing through AB through point B.  You may need to Edit>>Move Point a bit to bring points C and D closer. Get something that looks basically like the image to the right.  Draw a perpendicular to line AD through point D (which will give you a point E). Because alternate interior angles are equal (both 90), line AB and line DE are ______? * print or print screen / paste this image*  Also draw a perpendicular to line BC through point C (which will give you a point F). Because alternate interior angles are equal (both 90), line AB and line CF are ______?  Go to Edit >> Move Point, and slide points C and D around. You should be able to get things so that line DE and line CF intersect. This means that (1) you have two distinct lines through the same point (the point of intersection) that are parallel to a given line, and * print or print screen / paste this image* therefore (2) lines which are parallel to a third line are not necessarily ______?

Q4: Triangle area:

 Plot 3 points (A, B, and C).  Draw segments A to B, B to C, C to A.  Measure triangle ABC (gives all sides and all angles).  Record segment lengths: AB = BC = CA =

 Draw perpendicular to segment AB through C.  Measure distance CD CD =  Calculate value of .5(base)(height), using AB as the base, and CD as the height: .5(AB)(CD) =

 Draw perpendicular to segment BC through A.  Measure distance AE AE =  Calculate value of .5(base)(height), using BC as the base, and AE as the height: .5(BC)(AE) =

You should recognize .5(base)(height) as the formula for the area of a triangle in Euclidean geometry. Does it work as the area formula in hyperbolic geometry? Explain why/why not.

Q5: SAS congruence

 Start by constructing two segments of the same length: Constructions>> Draw Segment of Specific Length.  Then, with vertices at points A (on one segment) and C (on the other), Constructions>> Draw Ray at a Specific Angle (pick an angle). [ Click B to A to somewhere, then D to C to somewhere.]  By default, it should put point on those rays that are the same length as the original segments. (You can measure distance to check, and move point if needed.)  This means SAS = SAS by construction.  Form triangles by drawing segment EB and segment FD. * print or print screen / paste this image*  Measure triangle ABE and triangle CDF.  Are the triangles congruent?  Since ASA congruence can be proven from SAS * print or print screen / paste this image* congruence, we should have that as well. Construct a pair of triangles with ASA = ASA (construct a matching side for each, then construct matching pairs of angles at each endpoint). Measure. Congruent? (Hope so!)