Much of the study of geometry that we've done so far has consisted of defining terms and describing charateristics of various figures and their special cases. All of this study lays a foundation for one of the most important applications of geometry: proving shapes and figures are congruent. We've already discussed the congruence of segments and angles, but in the real world the congruence of regions in a plane is even more relevant. Since we can't easily prove the congruence of any region in the plane, we'll focus on simpler regions like those bound by polygons. And, like always, the study of polygons results in the study of triangles.
For two polygons to be congruent, they must have exactly the same size and shape. This means that their interior angles and sides must all be congruent. Not only must these parts be congruent, but they must be situated in a one-to- one correspondence, meaning each side in one polygon specifically corresponds to another side in the other polygon, and each pair of parts is congruent. To prove such a situation would be a tough task. That's why studying the congruence of triangles is so important--it allows us to draw conclusions about the congruence of polygons, too. We'll see how the six parts of a triangle correspond to one another, and how they must be aligned to signify congruence. We'll also study some techniques--shortcuts, really--to prove the congruence of triangles. We'll only work on informal proofs, the study of formal proofs in geometry will have to wait until the SparkNotes in Geometry 3. Finally, we'll take a look at similarity between triangles. Similarity is a lot like congruence, except it only requires the same shape, not size. After this section, we can focus on refining our skills for proving congruence. For now, we'll have to learn exactly what it means.
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