Chapter 5
5.1 The Polygon Angle-Sum Theorem
HOMEWORK:
Lesson 5.1/1-14
Objectives
• Define polygon, concave / convex
polygon, and regular polygon
• Find the sum of the measures of interior
angles of a polygon
2
Definition of polygon
• A polygon is a closed plane figure formed by
3 or more sides that are line segments;
– the segments only intersect at endpoints
– no adjacent sides are collinear
• Polygons are named using letters of
consecutive vertices
3
Concave and Convex Polygons
• A convex polygon has no
diagonal with points outside
the polygon
• A concave polygon has at
least one diagonal with points
outside the polygon
4
Regular Polygon Definition
• An equilateral polygon has all sides congruent
• An equiangular polygon has all angles
congruent
• A regular polygon is both equilateral and
equiangular
Note: A regular polygon is always convex
5
Polygon Angle-Sum Theorem
The sum of the measures of the angles of
an n-gon is
SUM = (n-2)180
ex: A pentagon
has 5 sides.
Sum = (n-2)180
Sum = (5-2)180
Sum = (3)180
Sum = 540
Sum of Interior Angles in Polygons
Convex Polygon # of
Sides
# of Triangles
from 1 Vertex
Sum of Interior Angle
Measures
Triangle
3
1
1* 180 = 180
Quadrilateral
4
2
2* 180 = 360
Pentagon
5
3
3* 180 = 540
Hexagon
6
4
4* 180 = 720
Heptagon
7
5
5* 180 = 900
Octagon
8
6
6* 180 = 1080
n-gon
n
n–2
(n – 2) * 180
7
Sum of Interior Angles
Find m∠ X
The sum of the measures of the
interior angles for a
quadrilateral is
(4 – 2) * 180 = 360
The marks in the illustration indicate that m∠X = m∠Y = x
So the sum of all four interior angles is
x + x + 100 + 90 = 360
2 x + 190 = 360
2 x = 170
m∠X = 85
8
Polygon Names
MEMORIZE THESE!
3 sides
4 sides
5 sides
6 sides
7 sides
8 sides
9 sides
10 sides
11 sides
12 sides
n sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
n-gon
Naming A Polygon
SIDES
A polygon is named by the number of_____.
ex: If a polygon has
5 sides, you use
___
5 letters.
___
Polygon ABCDE
Example #1
ABCDEF
CDEFAB
1. Name________.
concave
Is it concave or convex?__________
DEABC
ABCDE
2. Name ________
convex
Is it concave or convex?__________
1
2
Example #2
• Find the interior angle sum.
a. 13-gon
b. decagon
(n – 2) 180
(13 – 2) 180
(11) 180
1980˚
(n – 2) 180
(10 – 2) 180
(8) 180
1440˚
The Number of Sides
Use polygon SUM formula to find the number of
sides in a REGULAR or EQUIANGULAR
polygon
SUM = (n – 2) 180
1. Given (or calculate) the sum of the angles
2. Solve for n
Example #3
How many sides does each regular
polygon have if its interior angle sum is:
Sum is
a. 2700
b. 1080
given
2700 = (n – 2) 180
2700 = (n – 2)
180
15 = n – 2
17 = n
17-gon
1080 = (n – 2) 180
1080 = (n – 2)
180
6=n–2
8=n
Octagon
ONE angle in a Polygon
Use polygon SUM and the number of sides in a
REGULAR or EQUIANGULAR polygon to find
ONE angle
ONE = (n – 2) 180 = SUM
n
n
1. Given (or calculate) the sum of the angles
2. Solve for ONE
Example #5
Find y
First calculate pentagon sum
Pentagon sum = 540˚
540 = 5 y
540 = y
5
108˚ = y
Sum is
calculated
Example #6
Find x.
Hexagon sum = 720˚
one angle of an
equiangular hexagon
SUM = 720 = 120˚
6
6
120˚
x makes a linear pair with an
interior angle
x = 180˚ – 120˚ = 60˚
x = 60˚
Summary:
SUM of the Interior Angles of a Polygon
S = (n – 2) 180
One Interior Angle of a REGULAR Polygon
One = (n – 2) 180 = SUM
n
n
Example #7
Find x.
Heptagon sum = 900°
900 = x + 816
84 = x
132
100
155
142
167
+120
816