AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE TEACHER NETWORK TRIANGULAR
Investigate different triangles PQR where the angle QPR is 40o
choosing different sizes for the angles at Q and R (for example 90o
and 50o or 72o degrees and 68o).
Draw your own diagrams accurately and draw the internal bisectors
of the angles at Q and R to meet at S, as shown in the diagram.
What is the size of angle QSR?
Make you own conjecture about the size of angle QSR and prove it.
SOLUTION
Angle QSR is always 110o
From ΔPQR, as the angles of a triangle add up to 180o
2x + 2y = 180 – 40 = 140o
So x + y = 70o
From ΔSQR, as the angles of a triangle add up to 180o and, whatever
the values of x and y, x + y = 70o:
angle QSR = 180 – x - y = 110o
NOTES FOR TEACHERS
Why do this activity?
This activity gives learners practice in drawing accurate geometric constructions including internal bisectors
of the angles of a triangle. They should gain confidence from being able to make and prove their own
conjecture as it is actually very easy and only involves the fact that the angles of a triangle add up to 180o.
Intended Learning Objectives (Grade 9 and 10 )
To be able to construct geometric figures accurately, using compasses, ruler and protractor appropriately,
including bisecting angles of a triangle. To be able to make and prove conjectures using simple geometric
properties of triangles.
Possible approach
Simply give this problem to the class on paper or write it on the board and work on individually. Ask them
to read the question for themselves telling them that they need practice in doing this as it is what they have
to do in tests. Make sure that they have, at least one set between two of compasses, rulers and protractors.
It is possible to do this activity without doing the accurate geometrical constructions so you can adapt this
activity if you only plan for your learners to make the conjecture and prove it.
Lead a plenary session in which several learners explain the proof and write it on the board.
Key questions
What do you know about the angles PQR and PRQ?
What do you know about the angles SQR and SRQ?
Possible extension
Learners who find ‘Triangular’ easy could try: Can You Explain Why?
Possible support
This diagram shows learners how to bisect an angle using ruler
and compasses.
With centre V draw an arc and label the points A and B where
the arc cuts the arms of the angle.
With centre A draw an arc. With centre B and the same radius
draw an arc. These arcs cut at C. Join VC.
Then VC is the internal bisector of angle AVC, that is angles
CVA and CVB are equal.