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.
