Q If two sides and a non-included angle are the same in two triangles, is this a test for congruency? If not, can you explain why? The Sine rule would suggest that if you know two sides and the angle opposite one of them, you must be able to find another angle, giving the ASA, AAS reason for congruence.
A Shapes are congruent when they have the same size and shape. There are four ways to prove that two triangles are congruent:
* By showing they have matching sides (SSS or side-side-side).
In the diagram, the AB = EF = 11.3cm; BC = DF = 5cm; and AC = DE = 7.3cm, so the two triangles must be congruent. In notation, the three-line symbol (:) is used for congruence: DABC: DDEF. Pupils sometimes confuse this with three angles being the same. If there are three angles matching this does not show congruence, but it does mean that the two triangles are similar.
* By showing that two sides and the included angle (the angle between the two sides) match (SAS or side-angle-side). Here, BC = DF = 5cm and AC = DE 7.3cm; A CB (the angle between BC and ACE DF (the angle between DE and DF125x. So DABC: DDEF.
When I had your letter the first thing I did was to play with PowerPoint. A quick creation demonstrates that the triangles are only congruent if the angle is the included angle. I used “connectors” in the AutoShapes menu.
Create two lines of different lengths with the connectors and then a third.
If you increase the angle between the lines you can clearly see that the triangle changes and the line opposite the angle increases or decreases. If you have two sides the same and keep the angle between them constant, the triangle cannot be changed. A good interactive geometry package helps students engage with these rules. To answer part of your original question:No, you cannot use the sine rule as a proof because sin xx = sin (180x - xx).
* The third proof is showing two matching angles and a corresponding side (AAS, angle-angle-side). I describe “corresponding side” to pupils by saying that the pattern of angles on the side is the same. In this case, BC has the angle 125o on one end and an unmarked angle at the other end, matching DE and making them corresponding sides. B AC = D EF and BC = DF = 5cm, satisfying all the conditions. So DABC: DDEF.
* Lastly, two right-angled triangles are congruent when they have the same hypotenuse and another matching side.
On the internet, I found an excellent article in the Scottish Mathematical Council’s Journal by Professor Adam McBride from Strathclyde University about proofs, including the proofs relevant to congruent triangles: www-maths.mcs.st-andrews.ac.uk smcjournalmcb.html
There is still a lot of learning to be had from exploring rules associated with congruent triangles by getting pupils to investigate them using a ruler, pencil, and protractor. We assume that pupils in the higher sets are good at measuring accurately - but there is always room for practice!
I created the poem below to highlight the differences between congruent and similar in maths. This is available as a pdf for you to use with your classes at www.mathagonyaunt.co.uk
