21 Triangles
The most obvious fact about a triangle is that it is a figure with three sides and three
angles (tri-angles). Trigonometry is the theory which we use to ‘measure the triangle’,
whether it is the size of the angles, the length of the sides, or the enclosed area. This
shape – one of the simplest of all figures – is of enduring interest.
The triangle’s tale
There is a neat argument to show that the angles in any triangle add up to
two right angles or 180 degrees. Through the point or ‘vertex’ A of any triangle
draw a line MAN parallel to the base BC.
The angle A C which we’ll call x is equal to the angle BÂM because they are
alternate angles and MN and BC are parallel. The other two alternate angles are
equal to y. The angle about the point A is equal to 180 degrees (half of 360
degrees) and this is x + y + z which is the sum of the angles in the triangle. QED
as was often written at the end of his proofs. Of course we are assuming the
triangle is drawn on a flat surface like this flat piece of paper. The angles of a
triangle drawn on a ball (a spherical triangle) do not add up to 180 degrees but
that is another story.
Euclid proved many theorems about triangles, always making sure this was
done deductively. He showed, for example, that ‘in any triangle two sides taken
together in any manner are greater than the remaining one’. Nowadays this is