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Geometry: Polygons

Introduction to Triangles

Problems

Problems

Of all geometrical shapes, triangles are probably the most important. The most remarkable and important property of triangles is that any polygon can be split up into triangles simply by drawing diagonals of the polygon. This fact forms the basis for understanding why the interior angles of polygons add up to 180(n-2) degrees. The interior angles of a triangle always add up to 180 degrees. This can easily be proved by the congruence of alternate interior angles. From a given vertex of a polygon with n sides, (n-3) diagonals can be drawn. Every diagonal drawn from a single vertex of a polygon creates one triangle within the polygon, except for the last diagonal, which creates two triangles. For each triangle created within the polygon, 180 degrees of interior angles are created. (Of course the angles were there before the diagonals were drawn, but now they can be measured.) So n-4 diagonals of a polygon create one triangle each, and one diagonal, the last one to be drawn, creates two triangles. This means that n-2 triangles can be drawn into a given n-sided polygon. This is why the sum of all interior angles of an n-sided polygon is always 180(n-2) degrees. See the figure below for how the process looks.

Figure %: A polygon is divided into triangles, and the sum of its interior angles is shown to be 180(n-2) degrees.
The above polygon has n = 6 sides. n-3 = 3 diagonals can be drawn from a given vertex, yielding n-2 = 4 triangles. (n-2)180 = 720 degrees of interior angles in a 6-sided polygon.

This is only one way that triangles help demonstrate properties of polygons in general. There are many more. Triangles can be categorized many different ways, allowing us to focus on special characteristics of certain triangles that we can create within a polygon. This is the usefulness of triangles. For now, it's good just to know what they are. The Geometry 2 SparkNotes discuss all of the ways to use triangles.

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