1. The Polygon Angle-Sum Theorems

2. 3.4 The Polygon Angle-Sum Theorems
Polygon: a closed plane figure with at least three sides that are segments
Not a polygon;
Not enclosed
Not a polygon;
Two sides intersect
A polygon

3. Naming a Polygon Name a polygon by its vertices. A ABCDE or AEDCB
Start at one vertex and go around in order B E C D

4. Naming a Polygon Three polygons are pictured. Name each polygon: L P M

5. Classifying a Polygon by the number of sides:
Name 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon n
n-gon

6. Convex vs. Concave A Convex Polygon has all vertices pointing “out”
A Concave Polygon has one or more vertices
“caving in”

7. Classify
Classify each polygon by its sides. Identify each as convex or concave:
Hexagon; Convex, Octagon; Concave

8. Sum of Polygon Angle Measures
Use triangles to figure out the sum of the angles in each polygon:
# of Sides:
# of Triangles:
Total Degrees:
# of Sides:
# of Triangles:
Total Degrees:

9. Sum of Polygon Angle Measures
Number of Sides
Number of Triangles
Total Degrees inside Polygon 3 1
180 4 5 6 n

10. Theorem 3-9 Polygon Angle Sum Theorem
The sum of the measures of the angles in a polygon
is (n – 2)180.
Find the sum of the measure of the angles of a 15-gon.

11. Polygon Angle Sum
The sum of the measures of the angles of a given polygon is How many sides does the polygon have?
Use (n – 2)180 :

12. Using Polygon Angle-Sum Theorem
Find the measure of <Y in pentagon TVYMR at the right. R T
135°
Use (n – 2)180 M
90° Y V
Write an equation to solve for <Y

13. Using Polygon Angle-Sum Theorem
Pentagon ABCDE has 5 congruent angles. Find the measure of each angle.
Use the Polygon Angle-Sum Theorem: (n – 2)180
Divide the total number of degrees by the number of angles:

14. Exterior Angles What do you notice about each set of exterior angles?
80°
75°
115° 2 1
150°
99°
130°
71°
70°
88° 1:
86° 3 2: 3:
70°
46°

15. Theorem 3-10 Polygon Angle-Sum Theorem
The sum of one set of exterior angles for any polygon is 360°. 1 5 2 4 3
m<1 + m<2 + m<3 + m<4 + m<5 = 360°

16. Polygons Equilateral Polygon: all sides congruent
Equiangular Polygon: all angles congruent
Regular Polygon: all sides and all angles congruent (equiangular and equilateral)
*If a polygon is a regular polygon then all of the exterior angles are also congruent.