Properties of a Parallelogram

Using the definition given above, sketch a parallelogram. Find as many properties of this shape as you can by measuring sides, angles, diagonals and so on. Check that the properties you find are invariant when you sketch a different parallelogram.

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You can also explore parallelograms by opening the interactive file ‘2c Polygon Properties’. When you drag your parallelogram to different positions you may form some special paral- lelograms. What are they and what special properties do they have? Using the file you can also explore properties of other special quadrilaterals like trapeziums and kites. On each page of this file, the basic definition used to construct the shape is described.

Investigating parallelograms leads to a list of properties that probably include the fol- lowing: opposite sides are equal, opposite angles are equal, the diagonals bisect each other.There are two interesting follow-ups to the process of compiling such a list.The first is: how sure can you be that these properties hold true for all parallelograms? The list is the result of a set of confirming instances and is therefore based on inductive or

empirical reasoning. Even though dragging enables you to generate what seems like a

very large number of such instances, and which may seem extremely convincing, it does not alter the fact that only a finite number of cases have been tested. The con- clusions remain at the level of conjectures. However, using

your knowledge of congruent triangles, you could reason

deductively to justify conjectures, by making use of the defi-

nition of a parallelogram, and properties of parallel lines. For example, consider the diagram in Figure 2.2c.

Using properties of parallel lines and transversals, two

pairs of equal angles can be identified in the two triangles.This shows that the trian- gles are similar. But the two triangles also share a common corresponding side. Therefore, the condition ASA can be used to show that the triangles are in fact con- gruent, or you can reason that one is a scaled copy of the other, and a pair of corresponding sides is equal, so the scale factor must be 1. From this it follows that corresponding pairs of opposite sides of the parallelogram must always be equal.

In the same way, for example, by introducing the other diagonal to the drawing and creating more triangles, more properties can be deduced.The same process can be applied to other quadrilaterals. Trying to justify the generality of properties you have found by using deductive reasoning involves taking stated properties as the starting point, and using known and agreed facts to produce a chain of reasoning that ends up with the conjecture. This is reasoning on the basis of properties. If, for example, you start with the property that the diagonals bisect each other, you cannot suddenly say that opposite sides are parallel (because you know or believe the figure is a parallelogram). You have to use only the properties you start with to deduce consequences.

Another follow-up to this activity is to take one of the properties you have found and to ask yourself a question like the following: ‘If I draw (or make with geostrip) a quadrilateral with both pairs of opposite sides equal, will it be a parallelogram?’ If you find that it is indeed a parallelogram, then this property gives you a different way to construct the figure. Starting from different properties and trying to deduce others is very much like adopting different points of view, and then trying to see if they lead to the same conclusions.

Paulus Gerdus (1988) describes how Mozambican farmers construct the rectangu- lar base of their houses. Two ropes of equal length are knotted together at their mid-points. A bamboo stick whose length is the desired width of the house is laid down and two ends of the ropes are attached.The ropes are then pulled taut to deter- mine the other two vertices of the rectangle. (See Figure 2.2d.) This method is also used by carpenters in Europe.

LANGUAGE AND POINTS OF VIEW 25

Figure 2.2c

Clearly, the property that is being used here is that the diagonals of a rectangle are equal and bisect each other.The question is, if a quadrilateral is constructed with this property, does it ensure that the quadrilateral will be a rectangle? Investigate this using dynamic geometry software and also try to argue deductively. The key feature that you must demonstrate is that the angles are right angles.

It is a valuable exercise to find alternative constructions for well-known special polygons such as a square, rhombus or kite. When you try to construct a particular shape, you find that the concept of minimum conditions appears quite naturally. For example, you might have a definition of a square in your mind as a quadrilateral with all four sides equal and all four angles right angles. It is undoubtedly true that a square possesses all these properties. But when you construct it, do you actually need to use all these facts? In practice, you will find that the square is completed before you finish using them all. For example, the sequence in Figure 2.2e illustrates one possible way of constructing a square.

This may not be the ‘best’ way to construct a square (what could ‘best’ mean?) but it works. The point is that at each stage another known property of a square is being incorporated. Notice, too, that when the two perpendicular lines are drawn in the third stage, you could just as easily draw two lines that are parallel to the adjacent sides already drawn. When constructing using dynamic geometry software, you can check how successful your construction is by seeing if it is ‘robust’. That is, when you drag the independent elements of your construction, the shape will vary in size and orien- tation but the intended shape remains.

The process of finding alternative minimum conditions is known as characterising, because the different sets of conditions each characterise the shape being studied, in the sense of providing properties from which all the other properties can be deduced. 26 BLOCK 1