Why do this problem
utilises congruent triangles to provide a simple proof.
There are opportunities for discussion of "why it works" and for
drawing upon the historical context of angle trisection as well as
considering extending the idea to obtuse angles. You might wish to
use this problem after you have completed some work on congruence
tests (see notes below).
Making a carpenter's square and testing this out is a good first
Discussion of what mathematical tools learners might be able to
make use of can lead to ideas such as:
- identifying equal angles
- Possibilities of identifying congruent triangles.
After discussion of a range of possible tools and approaches,
ask learners to construct an argument and display it on a poster
for others to review.
One way to manage a review is to give pairs of learners two
sets of post-its (different colours). They can then go around the
room reading and evaluating other people's arguments and offer two
comments using their post-its. One comment should identify a good
aspect of the argument and another should identify where the
argument could be improved.
- Why might this work?
- What mathematical structures and ideas might you use if you
want to show angles are equal?
- Why can't you use this method as it is to trisect angles of
$90^o$ or more? Could you adapt it so that it could be used to
trisect an obtuse angle?
Can you extend this idea in some way to trisect an obtuse angle?
Use tracing paper to draw one of the congruent triangles and 'lay
it over' other sections of the diagram to help identify
the three congruent triangles.
Revisit congruence tests. See the notes below.
Use the images in this document
place the stages of the "construction" in order.
Historical notes on angle trisection:
Investigating congruence tests:
You will need rulers, protractors and pairs of compasses.
Ask everyone to construct a triangle with sides 5, 6 and 8cm using
rulers and compasses.
Cut them out or trace them.
Veryfy that everyone has drawn the same traingle whether it is
rotated, reflected or transposed in any way by overlaying images.
Examine counter examples.
With further examples establish that three sides uniquely defines a
Discuss the possiblility of other sets of three pieces of
information, what could they be?
Class then establishes minimum requirements to uniquely define a
triangle, possibly establishing that two pieces of information is
insufficient on the way.