Angle trisection
Problem
Angle Trisection printable sheet
A classical Greek problem was to find a way to trisect an angle using just a ruler and a pair of compasses. However, this is impossible.
It is possible though, to trisect an angle using a carpenter's square, as demonstrated in the video below.
Can you explain why this works?
Can you extend the idea to trisect an obtuse angle?
Getting Started
Student Solutions
AB is perpendicular to AO. AB, BC and CD are all equal in length.
Consider triangles OCB and DCO. Since OC is perpendicular to BD (as this is part of the carpenter's square), the two triangles are right-angled. They share a side (OC), and so since BC = CD, they are congruent. Hence angle BOC is equal to angle COD.
Now consider triangle OAB. By construction, AB is perpendicular to AO, so this triangle is also right-angled. It shares a hypotenuse with triangle BOC, and since AB = BC, OAB and OCB are congruent too. So angle AOB is equal to angles BOC and COD, and the angle has been trisected.
Teachers' Resources
Why do this problem
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).
Possible approach
Making a carpenter's square and testing this out is a good first step.
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.
Key Questions
- 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?
Possible support
Revisit congruence tests. See the notes below.
Use the images in this document to place the stages of the "construction" in order.
Possible extension
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.
Notes
Historical notes on angle trisection:
St AndrewsRutgers
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 triangle.
Discuss the possiblility of other sets of three pieces of information, what could they be?
AAA
ASA
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Class then establishes minimum requirements to uniquely define a triangle, possibly establishing that two pieces of information is insufficient on the way.