geometry-labs

compass and straightedge.You may also use other
construction tools, such as the Mira, patty paper
(see next paragraph), or computer software.This
lab, and most of the other construction labs, can
be readily adapted to those tools or to some
combination of them.
Patty paper is the paper used in some restaurants
to separate hamburger patties. It is inexpensive
and is available in restaurant supply stores and
from Key Curriculum Press. It is transparent,
can be written on, and leaves clear lines when
folded.These features make it ideal for use
in geometry class, as tracing and folding are
intuitively much more understandable than the
traditional compass and straightedge techniques.
In my classes, I tend to use patty paper as a
complement to a compass and straightedge,
rather than as a replacement. (I learned about
patty paper from Michael Serra. See his book
Patty Paper Geometry.)
In content, the lab focuses on the Triangle
Inequality. It is a useful preview of the concept of
congruent triangles as well as a good introduction
to the use of basic construction techniques.
Answers
1. See student work.
2. Answers will vary.
3. Sides a and b are too small, so they cannot
reach each other if e is the third side.
4. Possible: aaa, aab, abb, acc, add, ade, aee, bbb,
bbc, bcc, bcd, bdd, bde, bee, ccc, ccd, cce, cdd, cde,
cee, ddd, dde, dee, eee
Impossible: aac, aad, aae, abc, abd, abe, acd, ace,
bbd, bbe, bce
Discussion Answers
A. No. Since 2 4 does not add up to more
than 8.5, the two short sides would not
reach each other.
B. No. Since 2 1.5 does not add up to more
than 4, the two short sides would not reach
each other.
C. It must be greater than 2 and less than 6.
D. The two short sides add up to more than the
long side. Or: Any side is greater than the
difference of the other two and less than their
sum. (This is known as the Triangle Inequality.)
Lab 3.2: Triangles from Angles
Prerequisites: This lab assumes that you have
taught your students how to copy angles with
a compass and straightedge. As with the previous
lab, you may also use other construction tools,
such as the Mira, patty paper, or computer
software.This lab, and most of the other
construction labs, can be readily adapted to
those tools or to some combination of them.
This lab reviews the sum of the angles of a
triangle. It previews the concept of similar triangles
and introduces basic construction techniques.
If it is appropriate for your class at this time,
you may discuss the question of similar versus
congruent triangles.The triangles constructed in
this lab have a given shape (which is determined
by the angles), but not a given size, since no
side lengths are given.You could say that the
constructions in the previous lab were based
on SSS (three pairs of equal sides), while the
ones in this lab are based on AAA, or actually
AA (two—and therefore three—pairs of equal
angles). SSS guarantees congruent triangles:
All students will construct identical triangles.
AA guarantees similar triangles: All students
will construct triangles that have the same
shape, but not necessarily the same size.
Answers
1. See student work.
2. Answers will vary.
3. No.The sides would not intersect.
4. Possible: Triangles can be constructed using
the following pairs of angles: 1,1; 1,2; 1,3;
1,4; 1,5; 2,2; 2,3.
Impossible: Triangles cannot be constructed
using the following pairs of angles: 2,4; 2,5;
3,3; 3,4; 3,5; 4,4; 4,5; 5,5.
Geometry Labs Notes and Answers 177
© 1999 Henri Picciotto, www.MathEducationPage.org