Section 34: The Polygon AngleSum Theorems

1

• Section 3-4 The Polygon Angle-Sum Theorems
• A Polygon is a closed plane figure with at least
  three sides that are segments. The sides
  intersect only at their endpoints, and no
  adjacent sides are collinear.

A
E
B
Not A Polygon
C
D
A Polygon
2

• To name a polygon, start at any vertex and list
  the vertices consecutively in a clockwise or
  counterclockwise direction.

G
A
Two names could be AGDKPT and PTAGDK
Vertices D, K, P, T, A, G
D
T
Sides AG, GD, DK, KP, PT, TA
Angles ? A, ?G, ?D, ?K, ?P, ?T
K
P
3

• Student example Three polygons are pictured
  below. Name each polygon, its sides, and its
  angles.

Polygon ABE AB, BE, EA ?A, ?ABE, ?BEA
B
C
Polygon BCDE BC, CD, DE, EB ?C, ? D, ?CBE, ?BED
A
D
E
Polygon ABCDE AB, AE, BC, CD, DE ? C, ? D, ? A,?
ABC, ?AED
4

• You can classify a polygon by the number of
  sides it has. The table shows the names of some
  common polygons.

5

• Convex Polygon has no diagonal with points
  outside the polygon.
• Concave Polygon has at least one diagonal with
  points outside the polygon.

6

• Theorem 3-9 Polygon Angle-Sum Theorem, The sum
  of the measures of the angles of an n-gon is
  (n-2)180.
• Example Find the sum of the measures of the
  angles of a 15-gon.
• Sum (n-2)180
• (15-2)180
• (13)180
• 2340

7

• Student example Find the sum of the measures of
  the angles of a 13-gon.
• Student example The sum of the measures of the
  angles of a given polygon is 720. Find the
  number of sides of the polygon.

1980
6 sides
8

• Theorem 3-10 Polygon Exterior Angle-Sum Theorem,
  The sum of the measures of the exterior angles of
  a polygon, one at each vertex is 360.

1
5
2
4
3
9

• Equilateral Polygon has all sides congruent.
• Equiangular Polygon has all angles congruent.
• Regular Polygon is both equilateral and
  equiangular.

10

• Class work
• Page 147-148
• Questions 1, 5, 16, 17, 32, 40
• Homework
• Page 147-149
• Questions 3-4, 6-9, 12-14, 18-20, 22-24, 26-27,
  34-36, 41-42, 55, 56