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Assorted Theorems

Throughout our study of geometry in Geometry1 and the first three SparkNotes of the Geometry2 series, we've essentially built up a library of knowledge about the kinds of figures that compose geometry. These include the building blocks, various constructions, and shapes like polygons and circles. In addition, we've looked at three-dimensional surfaces and the different ways to measure polygons. The last topic dealt with the concepts of congruence and similarity and the consequences inherent when triangles or certain parts of triangles are congruent or similar. In congruence, we looked at the techniques for proving that the triangle as a whole was either congruent or similar. A major part of doing so, we learned, involves analyzing a figure and determining which parts, if any, are either congruent, proportional, or neither. Only then, when enough is known about certain parts, can one of the techniques for proving congruence be used. We already know a few of these methods. For example, we know that a perpendicular bisector is perpendicular to the segment it bisects, and intersects that segment at its midpoint. This fact, along with the others we have learned, are related in that they are all true by definition.

In the following SparkNote, we'll learn some of the more complex relationships between parts of figures. These facts are known as theorems. The basic theorems that we'll learn have been proven in the past. The proofs for all of them would be far beyond the scope of this text, so we'll just accept them as true without showing their proof. Eventually we'll develop a bank of knowledge, or a familiarity with these theorems, which will allow us to prove things on our own. After the following lessons, we'll recap everything we know about certain shapes, every relationship between parts, every fact that is true by definition--everything in our knowledge bank of figure analysis. With the tools you already have learned, along with those you're about to learn, you'll be able to conclude a surprisingly great amount from a figure about which you were told very little. This process of proving statements geometrically is one of the most important goals of geometry. For now, we'll only prove things informally. In the SparkNotes making up Geometry3 we'll learn how to do formal proofs.