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Geometry: Section 7.3
Name:
Objective: To learn how to find the number of diagonals and the sum of the interior and of the
exterior angles of a polygon.
Review: Polygon is the union of segments in which no two
consecutive segments are collinear and the segments intersect
exactly two other segments, one at each endpoint.
Types of polygons:
~ can be classified according to angles
Convex – all interior angles are acute or right
Concave: At least one interior angle is obtuse
~ can be classified according to the number of sides
# of sides
3
Name
Triangle
polygons
4
5
6
7
8
9
10
12
15
pentadecagon
n
Theorem 7.7: The sum of the interior angles of a polygon with n
sides is given by the formula: Si = (n ! 2)i180 .
This is proved by drawing all the diagonals from a vertex, and
adding the angles of each of these nonoverlapping triangles.
If the sum of the interior angles of an n-gon is 1800, how many sides must if have?
Theorem 7.8: If one exterior angle is taken at each vertex,
the sum Se of the measures of the exterior angles of a
polygon is given by Se = 360.
Proof: If the sum of the n interior angles is (n-2) 180
and we have n sets of supplementary interior and exterior angles,
then the sum of these interior and exterior angles must be 180.
Interior angles + exterior angles = 180 n
But interior angles Si = (n ! 2)i180 , so
(n ! 2)i180 + sum of exterior angles = 180 n.
...
Theorem 7.9: The number of diagonals that can be drawn in
a polygon of n sides is given by the formula:
d=
n ( n ! 3)
2
This can be proved using mathematical induction, which we have not yet learned, so
we will confirm this for the first few types of polygons and then accept it for now.
# of Sides
3
Name
figure
# of diagonals drawn
Triangle
4
5
6
7
Homework: 7.3 pgs. 309-312 #1, 3, 5, 6a, ,c, 10a, c, 15, 16, 20
d=
n ( n ! 3)
2