Finally the foundation is laid to begin to look at some of the basic shapes that characterize geometry. Most of the familiar shapes, such as squares, triangles, rectangles, are part of a larger subset of shapes called polygons. Polygons are geometric figures that consist of the union of segments attached at their endpoints, ending at the point at which they started, and not intersecting each other. This definition is messy, but after seeing some polygons, it will be much easier to understand. Basically, polygons have straight sides and enclose a region in the plane. Because they are defined in fairly specific terms, polygons are governed by useful sets of rules that allow us, the geometry scholars of the world, to infer things about them that aren't stated in the problems that confront us. These special properties come in handy later on when we'll try to prove things geometrically.

Polygons can be classified according to the number of sides they have, their interior angles, the length of their sides, or all three. For example, A four-sided polygon whose angles and sides are all congruent is a square. After laying out the general rules of polygons, we'll take a brief look at some important types of polygons. Quadrilaterals are classified by the relationships that exist between their sides. Special quadrilaterals like parallelograms and trapezoids have properties that make it easy to analyze any quadrilateral. Another even more important polygon is the triangle. A triangle is especially useful in the real world because by themselves triangles are simple, but they can be united to compose any polygon in the world. With a foundation of knowledge about triangles on the individual level, there is no limit to what information you can extract about a shape simply by dividing it into multiple triangles. Before we get ahead of ourselves, though, we have to understand exactly how a point or a line becomes a recognizable shape.

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