1. 5.1 The Polygon Angle-Sum Theorem
Chapter 5
5.1 The Polygon Angle-Sum Theorem
HOMEWORK:
Lesson 5.1/1-14

2. Objectives
Define polygon, concave / convex polygon, and regular polygon
Find the sum of the measures of interior angles of a polygon

3. 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

4. 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

5. 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

6. 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.

7. 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* 180 = 540
Hexagon 6
4* 180 = 720
Heptagon 7
5* 180 = 900
Octagon 8
6* 180 = 1080
n-gon n
n – 2
(n – 2) * 180

8. Sum of Interior Angles Find m∠ X m∠X = 85
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 = 360
2 x = 360
2 x = 170
m∠X = 85

9. Polygon Names MEMORIZE THESE!
3 sides Triangle
4 sides Quadrilateral
5 sides Pentagon
6 sides Hexagon
7 sides Heptagon
8 sides Octagon
9 sides Nonagon
10 sides Decagon
11 sides Undecagon
12 sides Dodecagon
n sides n-gon

10. Naming A Polygon A polygon is named by the number of_____.
ex: If a polygon has
___ sides, you use
___ letters.
Polygon ABCDE
SIDES 5 5

11. Example #1 1. Name________. 2. Name ________
Is it concave or convex?__________
2. Name ________
CDEFAB
ABCDEF
concave
DEABC
ABCDE
convex 1 2

12. Example #2 Find the interior angle sum. a. 13-gon b. decagon
(13 – 2) 180
(11) 180
1980˚
(n – 2) 180
(10 – 2) 180
(8) 180
1440˚

13. The Number of Sides
Use polygon SUM formula to find the number of sides in a REGULAR or EQUIANGULAR polygon
SUM = (n – 2) 180
Given (or calculate) the sum of the angles
Solve for n

14. Example #3
How many sides does each regular polygon have if its interior angle sum is:
a  b 
Sum is 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

15. 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
Given (or calculate) the sum of the angles
Solve for ONE

16. Example #5 Find y 108˚ = y First calculate pentagon sum
Sum is calculated
540 = 5 y
540 = y 5
108˚ = y

17. Example #6 x = 60˚ Find x. x = 180˚ – 120˚ = 60˚ Hexagon sum = 720˚
one angle of an equiangular hexagon
SUM = 720 = 120˚
120˚
x makes a linear pair with an interior angle
x = 60˚
x = 180˚ – 120˚ = 60˚

18. 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

19. Example #7 Find x. 84 = x 900 = x + 816 Heptagon sum = 900° 132 100
155
142
167
+120
816
84 = x