SECTION 8.2 TWO-DIMENSIONAL FIGURES - Cengage Learning

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■ Mathematics ■
If you thought all mathematics
proofs were like the twocolumn
proofs in high school
geometry, you were mistaken.
Many, many mathematical
proofs are fun and based on
common sense that is
accessible to all people!
FIGURE 8.97
FIGURE 8.95
This is one of those cases where mathematics is counterintuitive. An exterior
angle is an angle formed by a side of the polygon and the extension of the
side adjacent to that side. Now go back and see whether you can draw the exterior
angles for the two polygons. How many exterior angles does a polygon
have? Then, and only then, look below. . . .
C
W
FIGURE 8.96
O
Section 8.2 / Two-Dimensional Figures 535
The diagrams in Figure 8.96 illustrate the exterior angles. As you may have
guessed, there is an exterior angle associated with each interior angle.
The sum of the interior angles of a polygon is 180n 2. What do you
think is the sum of the exterior angles of a polygon? . . .
Amazingly, the sum of the exterior angles of any polygon is 360 degrees.
You can see the proof for yourself by placing a pencil on vertex C of the triangle
above. Now “walk” the pencil around the triangle. That is, move it to vertex O,
now rotate the pencil so you are pointing toward W; note that the amount of
turn is exactly the exterior angle at O. Now walk the pencil to W. Turn the pencil
at W so you are pointing toward C; again, the amount of turn is equal to the
exterior angle at W. Finally walk to C and turn to face O. You find that the pencil
has made one complete turn. Do the same with quadrilateral PONY. Once
again, you find you make one complete turn.
All polygons have interior and exterior angles. When we focus on regular
polygons, there is an additional kind of angle. A central angle of a regular polygon
consists of a vertex at the center of the polygon and the two sides connecting
the center to adjacent vertices of the polygon (see Figure 8.97). The size of
the central angle is not difficult to determine if you recall that one complete
turn around a point is equal to 360 degrees. The measure of the central angle of
a regular n-gon (a polygon with n sides) is equal to 360n degrees. This and
other theorems developed in this section are used to advance our understanding
of the structure of geometry, which is then used by people in various fields.
For example, graphic artists use their sense of geometric structure to make very
beautiful and intricate designs, both on everyday objects like fabric and posters
and as pieces of art.
P
Y
O
N