Download Things to do in the program/applet Non

Things to do in the program/applet Non-Euclid
1. Start a new construction (File->New). Make a two lines (Constructions -> Draw Line). These
will probably be line AB and CD . Now switch to point-dragging mode (Edit->Move Point).
Drag point D around. Observe the many lines that go through C and do not intersect AB . All of
these are parallel. Choose the best word to complete the sentence:
a. In the Poincare disk, given a line and a point not on the line there are (no/exactly one/at
least one/more than one) line(s) through the point, parallel to the given line.
b. In Euclidean Geometry, if you have a line l and two lines m and n that are parallel to l
(parallel means do not intersect), then m and n are parallel to each other. Is this true in
the Poincare disk? (Try to find an example that shows it is not true)
2. Make a triangle (Constructions->Draw Line Segment). Measure the triangle. (Measure>Measure Triangle, then click the 3 vertices of the triangle). Switch to point dragging mode
(Edit->Move Point), and move the vertices around:
a. How small can you make the angle sum? What does the triangle look like when its
angle sum is small?
b. What does the triangle look like when its angle sum is large?
Drag the vertices until you have an interesting triangle
c. Construct another triangle by making a side, angle and side with the same
measurements as your interesting triangle. (Use Constructions-> Draw Segment
of Specific Length, and Constructions-> Draw Ray at Specific Angle). Now
measure this new triangle. Is it congruent to the first one? (It is congruent if its
measurements are the same—I would say that only the first two decimal places of
lengths were really trustworthy, and only the whole number part of an angle)
d. Construct another triangle by making ASA the same. Is the triangle congruent?
e. Construct another triangle with AA the same, with the length of the included
side being dragable. If you drag the segment so that AAS matches the first
triangle, does that make the triangles congruent?
f. If you drag the segment of your AA triangle, is the other angle always the
same? If not, when AAA matches, are the triangles congruent?
3. There are no rectangles in the Poincare disk. Make a quadrilateral with 3 right angles. What
is the fourth angle like?
4. A mathematician named Sacchieri came up with some rectangle-like things when trying to
prove that the parallel postulate wasn’t needed. A Sacchieri quadrilateral ABCD has right angles
at B and C, and sides AB and CD are congruent. Make a Sacchierri quadrilateral.
a. What property is/seems to be true of the angles at A and D?
b. What property is/seems to be true of the diagonals of ABCD?
5. The defining trait of a parallelogram is that its opposite sides are parallel: that is, when you
extend the opposite sides to lines, they do not intersect. Make a weird-looking parallelogram in
the Poincare disk and sketch it here:
6. A rhombus is a quadrilateral whose sides are all congruent. You can construct one this way:
 Draw a segment AB
 Draw a circle with center at A and radius out to B.
 Draw another segment AC where C lies on the circle.
 Draw circles: Center at B, radius to A and Center at C, and radius to A.
 Find the intersection of the circles D. BACD is a rhombus
a. Are the sides parallel?
b. Are the diagonals perpendicular?
c. Do the diagonals bisect each other?
d. Do the diagonals bisect the angles?
e. Are opposite angles congruent?