3.4 The Polygon Angle-Sum Theorems

Chapter 3: Parallel and Perpendicular Lines

3.4 The Polygon Angle-Sum Theorems

Polygon: a closed plane figure with at least three sides that are segments

A polygon Not a polygon;Not enclosed

Not a polygon;Two sides intersect

Naming a Polygon

Name a polygon by its vertices.

A

B

CD

E

ABCDE or AEDCB

Start at one vertex and go around in order

Naming a Polygon

Three polygons are pictured. Name each polygon:

L

M

NO

P

Classifying a Polygon by the number of sides:

Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

Convex vs. Concave

A Convex Polygon has all vertices pointing “out”

A Concave Polygon has one or more vertices

“caving in”

Classify

Classify each polygon by its sides. Identify each as convex or concave:

Hexagon; Convex Octagon; Concave

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:

Sum of Polygon Angle Measures

Number of Sides Number of Triangles

Total Degrees inside Polygon

3 1 180

4

5

6

n

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.

Polygon Angle Sum

The sum of the measures of the angles of a given polygon is 720. How many sides does the polygon have?

Use (n – 2)180 :

Using Polygon Angle-Sum Theorem

Find the measure of <Y in pentagon TVYMR at the right.

135°

M

Y V

TR

90°

Use (n – 2)180

Write an equation to solve for <Y

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:

Exterior Angles

What do you notice about each set of exterior angles?

130°

150°

80°

115°

71°

75°

99°

86°

70°

88°

70°46°

1 2

31:

2:

3:

Theorem 3-10 Polygon Angle-Sum Theorem

The sum of one set of exterior angles for any polygon is 360°.

1

2

3

4

5

m<1 + m<2 + m<3 + m<4 + m<5 = 360°

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.

Homework

Pg 147 1-25, 40-44, 47-49