Download page 1 of 2 Math 330A (Barsamian) Computer Project 4: The CACS

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Math 330A (Barsamian) Computer Project 4: The CACS Theorem
Concept Review: Conditional Statements and their Converses
Remember from Computer Lab 3 the definition of the converse of a conditional statement:
in words in symbols
Statement S: If Pthen Q
PQ
Converse of Statement S: If Qthen P
QP
Remember also that Statement S and the Converse of Statement S do not mean the same thing. They are
not logically equivalent. The fact that some conditional statement S is true or false has no bearing at all
on the issue of whether the converse of S is true or false. For example, in Computer Lab 3, we explored
the truth of the following statement and its converse.
Statement S:
Converse of S:
If a pair of alternate interior angles is
congruent, then lines L and M are parallel.
If lines L and M are parallel, then a pair of
alternate interior angles is congruent.
Euclidean Poincare Disk
(GSP)
(NonEuclid)
True
True
True
False
It was no surprise that Statement S should seem to be true in both our GSP drawings and our NonEuclid
drawings, because we had already proven in calss that Statement S is in fact a theorem of Neutral
Geometry. But that theorem tells us nothing about the truth of the converse of Statement S. It was
noteable, then, to find that the converse of S seems to be true in our GSP drawings but not in our
NonEuclid drawings.
The CACS Theorem
Today, we will be making drawings to illustrate the following conditional statement and its converse.
Statement R: If two sides of a triangle are congruent, then the two opposite angles are congruent.
Converse of R: If two angles of a triangle are congruent, then the two opposite sides are congruent.
Abbreviated version:
Statement R: congruent sides congruent angles
Converse of R: congruent angles congruent sides
We proved in class that both Statement R and the Converse of Statement R are always true in Neutral
Geometry. That is, they are both theorems of Neutral Geometry. Statement R is called the Isosceles
Triangle Theorem. (Remember the naming convention: Theorems are named for things in their
hypothesis, not for things in their conclusion.) But I find it simpler to think of the pair of theorems as
one big theorem called CACS:
CACS: Congruent Angles are always opposite Congruent Sides.
In this lab, I won’t give you explicit instructions about exactly what steps to do. Instead, I would like for
you to work together in groups of 3 or more students to figure out how to do each task. (Maybe the
whole class can have a discussion?) But I would still like each of you to make your own drawings.
Math 330A Computer Project 4
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Part I: Drawings to illustrate Statement R: Congruent Sides Congruent Angles
Geometer’s Sketchpad Tasks (Drawings of Euclidean Geometry)
1. Create an “adjustable” triangle that will have two congruent sides regardless of how the points
are moved around. (Hint: Consider the task of constructing a triangle △ ABC where AB ≅ AC .
Remember that in our drawings, line segment congruence means that the line segments have the
same length. What geometric shape could be used as an anchor for the points?)
2. Measure segments AB and AC and angles ∡C and ∡B .
3. Confirm that when you move the points of your drawing, the various parts automatically move in
a way that the lengths of segments AB and AC remain equal and the measures of angles ∡C
and ∡B remain equal.
4. Call me over to your computer to see your drawing. I will want to try moving the points around.
NonEuclid Tasks (Drawings of the Poincare Disk)
Same steps 1-4 from above. Again, call me over to your computer to see your drawing when it is
done. As before, I will want to try moving the points around.
Part II: Drawings to illustrate the Converse of R: Congruent Angles Congruent Sides
Geometer’s Sketchpad Tasks (Drawings of Euclidean Geometry)
1. Create an “adjustable” triangle that will have two congruent angles regardless of how the points
are moved around. (Hint: Construct a segment AB that will be the base of the triangle. Construct
a ray AC . This will create an angle ∡BAC . You would like to create another angle, this time at
point B, that has the same measure as angle ∡BAC . Use the technique that you learned in
Computer Lab 3 that involved constructing a new point by creating a rotated copy of some other
point. In this case, you will want to create a new D by rotating point A around point B by an
amount equal to the measure of angle ∡BAC . Once you get point D, construct ray BD . Finally
construct a point at the intersection of rays AC and BD .)
2. Measure angles ∡A and ∡B and the lengths of the two sides.
3. Confirm that when you move the points of your drawing, the various parts automatically move in
a way that the measures of angles ∡A and ∡B remain equal and lengths of segments AB and
AC remain equal.
4. Call me over to your computer to see your drawing.
NonEuclid Tasks (Drawings of the Poincare Disk)
Same steps 1-4 from above, but with a change in step 1. You cannot create a congruent copy of
an angle in NonEuclid in the same way that you constructed a copy in GSP. But remember that
in Computer Lab 3, when you needed to construct a congruent copy of an angle in the NonEuclid
program, you did it by reflecting a ray or a line across some other line that is the axis of
reflection. As in Computer Lab 3, you will need to create at least one additional line to use as a
guide or as a axis of reflection. When you get the triangle you need, hide the objects that were
created just to serve as guides or axes of reflections.When you are done, call me over to your
computer to see your drawing.