1. Section 3-4 Polygon Angle-SumTheorems
Name Polygons
Calculate Interior Angles
Calculate Exterior Angles
Understand Diagonals

2. The Basics . . .
Definition: 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.
Not Polygons
Polygons

3. Naming Polygons
Start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction. D
DIANE
IANED
ANEDI
NEDIA
EDIAN
DENAI
ENAID
NAIDE
AIDEN
IDENA E I N A

4. Definition: A diagonal of a polygon is a segment that connects two nonconsecutive vertices.

5. The Basics . . .
Definition: A convex polygon has all diagonals on the interior of the polygon.
Definition: A concave polygon has a diagonal on the exterior of the polygon.
Convex
Concave

6. Naming Polygons # Sides Polygon 3 triangle 4 5 quadrilateral 6 78 9 10 12 n
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
n-gon

8. Real-Life ConnectionsH
Benzene

9. Apply What You Already Know:
Theorem: The sum of the measures of the three angles in a triangle is 180 degrees.
How About a Quadrilateral? What is the sum of the angles? Explain your answer?

10. Polygons and Interior Angles Investigation Part I
Theorem: The sum Si of the measures of the angles of a polygon with n sides is given by the formula: Si = (n – 2)180.
Huh?
The polygon is divided into triangles.
The number of triangles is always two less than the number of polygon sides.
There are 180 in each triangle.

11. Polygons and Exterior Angles Investigation Part II
Theorem: 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 the formula
Se = 360
Huh?
If you cut out each exterior angle and arranged the vertices on top of one another they would form a circle of sorts.
exterior angle

12. More About Interior Angles Investigation Part III
180n – 360 = 180(n – 2)
Huh?
If you add all the angles of each triangle, then you include all the angles that go completely around the selected point. Hence, you need to subtract 360!

13. Diagonals of Polygons Investigation Part III
Theorem: The number, d, of diagonals that can be drawn in a polygon of n sides is given by the formula:
Huh?
From each of the n vertices you can draw n – 3 diagonals. Thus, there are n(n-3) diagonals total. But, by this method, each diagonal is drawn twice, so you must divide by 2.

14. Regular Polygons
Definition: A regular polygon is a polygon that is both equilateral and equiangular.
Theorem: The measure E of each exterior angle of an equiangular polygon of n sides is given by the formula
exterior angle