Quandaries and Queries


Name: tania
Who is asking: Student
Level of the question: Middle

Question: how do you prove that a rhombus' diagonals are perpendicular using the 2 column proof method?



The perpendicularity is visually evident if you recognize that the two diagonals are mirrors of the shape: one mirror diagonal takes the other diagonal onto itself (splitting the 180 degree straight angle at the crossing point). Unfortunately, this observation is probably outside the boundaries of 'two column proofs'. It is, of course, valid mathematical reasoning (sigh) and is a good place to start.

So you need to find a way to bring the reasoning above into the two column format.

This will involve generating pairs of isosceles triangles. If you orient the rhombus as a diamond, take the top pair of equal edges with the horizontal diagonal form a isosceles triangle. The bottom pair also make an isosceles triangle with the shared horizontal diagonal.

There are several steps to write down here, depending on what you already have 'in hand'. Let me outline one route that is closer to the 'reflections' above.

The top triangle, and the bottom triangle, have three corresponding sides equal. The top right and the bottom right (same length - rhombus). The top left and the bottom left (same length, rhombus). The shared horizontal diagonal. Therefore these triangles are congruent (SSS). What is the congruence (transformation) of one onto the other? Since the horizontal diagonal is unmoved, it must be a reflection in this fixed line.

The reflection takes the top vertex to the bottom vertex, and the vertical diagonal to the vertical diagonal (reversed) keeping the crossing point of the horizontal diagonal unmoved. In this reflection, the right top angle of the crossing of the diagonals goes onto the bottom right angle of the crossing. This shows these angles are equal. Since the vertical diagonal is a straight line, these two angles must be right angles (half a straight line angle). This is the required fact.

Notice that the same proof works for a kite (on its 'side' with the tip and tail on the horizontal diagonal). It is always nice when a proof covers extra examples - so you learned more from doing the proof!

In any two column proof, there is some set of 'accepted steps' which the text, or the classroom teacher, has established. These are not the same in every classroom (they are 'conventions': agreed by the group, for the section or course). So what is known and what is to be proven depends on this established set (sometimes called axioms, or prior results or ... ). Unfortunately, it is rare that this set has explicit references to facts about transformations (e.g. the reflection above). This is too bad, because many people actually reason with transformations, and we should be included among those doing mathematical reasoning.

I hope the text / teacher have agreed that 'congruence' really means a plane, distance preserving transformation: something that moves one object on top of another. If, by chance, they made the mistake of defining congruence as some abstract 'equality of properties, then they have missed the core concept. If they made that mistake, I cannot predict which rules they will accept or will reject in the two column proof.

Modern geometry (as we use it in applications and higher mathematics) does use transformations extensively, and does treat congruence as a transformation. I hope your classroom is at least open to that approach. Work with transformation does not really use the formal two column system, as that often gets in the way of developing the more important skill of 'geometric reasoning'.

Walter Whiteley