Angle trisection
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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
The diagram below shows the last frame from the video, with some lines and labels added. The thin grey line is parallel to AO and its perpendicular distance from AO is BC. The argument involves triangles that have the blue lines as sides.
You might find it helpful to annotate a copy of this diagram as you read the following argument.
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
There are a lot of important ideas in the video so taking time to explore these may support later reasoning.
You could begin by asking students to watch the video several times. You could slow down the video to give students opportunities to note what they have seen. After showing the video two or three times, you could also ask students to talk in pairs about what the video showed from start to end.
To begin exploring the reasoning behind the video and to engage further with how the carpenter's square is positioned, a good step could be to make a carpenter's square out of squared paper or card and test out the approach. You could organise this either with one carpenter's square in a full class discussion or with students working in small groups. Can students always position a carpenter's square as it is positioned in the video?
Now you can shift the focus to why the arrangement in the video trisects the given angle.
Discussing what mathematical tools learners might be able to make use of can lead to ideas such as:
- identifying equal angles
- identifying properties of the triangles (e.g. right angles, shared sides, congruence conditions).
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 $90^\circ$ 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. Encourage students to talk out loud as they do this or annotate the diagrams with what they notice in each diagram and how the diagrams are connected.
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:
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 possibility of other sets of three pieces of information, what could they be?
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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.