It may seem surprising to you, but being able to prove two triangles are congruent will now allow us to discover and prove many geometric properties, not only about triangles but about other figures as well! In this chapter, you will discover many properties of isosceles triangles, equilateral triangles, right triangles, and of triangles in general. Let us begin the discoveries by looking at the symmetries of triangles, and see where this leads us. Let's begin with the most symmetrical triangle of all: the equilateral triangle.
If you construct or draw an equilateral triangle, you can do some interesting symmetry experiments by simply folding the triangle. Begin by folding any vertex over onto the other vertex, as in the sequential steps below:
If you did this by actually folding an accurately constructed equilateral triangle made of paper, you probably noticed that the triangle is perfectly symmetrical, with reflection symmetry. Unfold the triangle and then fold it again, folding any vertex onto any other vertex. What seems to be true about the equilateral triangle? This symmetry will tell us a number of properties of the equilateral triangle, properties of its sides, its angles, the medians, altitudes and angle bisectors. You will find that in an equilateral triangle, each altitude from a point to the opposite side is congruent to each of the other altidudes, and this is also true for each median and angle bisector. As a matter of fact, each altitude in an equilateral triangle is also a median and an angle bisector! This is certainly not true for a "scalene" triangle (a triangle with NO congruent sides).
What would happen if you tried this with an isosceles triangle (one that was definitely not equilateral)? Construct or draw an isosceles triangle. Then try the folds shown below:
What conclusions can you make about the symmetry of an isosceles triangle? Does it have the same reflection symmetries as the equilateral triangle? What might this mean in relation to the angles? The altitudes, medians and angle bisectors? You will discover the most important property of an isosceles triangle: that it always has 2 congruent angles. The angle that are congruent are the angles "opposite" the congruent sides.In the drawing above, angle B is opposite side AC, angle A is opposite side BC. Since side AC is congruent to side BC, angle B is congruent to angle A. This is called " The Isosceles Triangle Theorem".
Paper folding activities, as we did above, are interesting, and they can result in beautiful works of art and clever three-dimensional shapes. Have you ever heard of Origami, the Japanese art of paper-folding? There are some fascinating web sites on this topic. The beautiful swan below came from a web page by an origami master.
And here is an origami crane:
To visit this web site, click on the link below:
Here's another very interesting website, with many links to other Origami sites:
Fold an origami cricket, following the steps below. Notice that these directions use many geometric terms. (Most origami instructions do not use mathematical terms!) This origami cricket will serve as an example of the kind of thing you will be doing as a project for this chapter.
Your Project assignment:
Fold a square piece of colored, good quality paper, to create an interesting Origami. It can be anything you like: an animal, a bird, or just an abstract three-dimensional shape. Then, using The Geometer's Sketchpad, or a compass and straightedge, create a step-by-step set of instructions that someone else could follow. You may look at the step-by-step examples that you saw on the web pages, but use the Sketchpad "cricket" example above to guide you, using geometric terms to describe the steps, although your instructions do not have to be as complicated as the cricket example. Letter the vertices of congruent triangles, and list them by groups as shown in the cricket example above.
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