### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

### Encircling

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

### Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

# Angle Trisection

### 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.
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.

### 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 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.

### Possible support

Revisit congruence tests. See the notes below.
Use the images in this document to place the stages of the "construction" in order.

### Notes

St Andrews
Rutgers

#### 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
...
Class then establishes minimum requirements to uniquely define a triangle, possibly establishing that two pieces of information is insufficient on the way.