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124
Consider a
(k
+
I)gon. Let
P
be a vertex and
Q,
R
the vertices
adjacent to
P.
Join
QR
as shown on the right. Since
k
?: 3, the line
Q R
separates the figure into two distinct pieces, triangle
PQ R
and
the
kgon
fonned by
QR
and all sides of the original
(k
+
I)gon
except
PQ
and
P R.
By the induction hypothesis, the sum of the
interior angles of this new
kgon
equals
(k
 2) 180
0
•
Also, triangle
PQR
has sum of interior angles equal to 180
0
•
Solutions to Exercises
The sum of the angles of the
(k
+
1 )gon is the sum of the angles in the
kgon
and in the triangle, that
is,
(k

2) 180
0
+
180
0
=
(k

1) 180
0
,
as desired. By the Principle of Mathematical Induction, the
result is true for all
n
?: 3.
29. We prove this by induction on
n,
the number of straight lines drawn. If
n
=
1, we have two regions
which, if colored with different colors, gives a proper coloring using just two colors. (By "proper," we
mean that bordering countries have different colors.) Now suppose that
k
?: 1 and the statement is
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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