Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.
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Transcript of Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.
![Page 1: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/1.jpg)
Chapter 5
5.1 The Polygon Angle-Sum Theorem
HOMEWORK:Lesson 5.1/1-14
![Page 2: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/2.jpg)
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Objectives
• Define polygon, concave / convex polygon, and regular polygon
• Find the sum of the measures of interior angles of a polygon
![Page 3: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/3.jpg)
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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
![Page 4: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/4.jpg)
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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
![Page 5: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/5.jpg)
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Regular Polygon Definition
• An equilateral polygon has all sides congruent• An equiangular polygon has all angles
congruent• A regular polygon is both equilateral and
equiangularNote: A regular polygon is always convex
![Page 6: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/7.jpg)
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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
![Page 8: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/8.jpg)
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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 = xSo the sum of all four interior angles is
x + x + 100 + 90 = 3602 x + 190 = 3602 x = 170
m∠X = 85
![Page 9: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/9.jpg)
Polygon NamesMEMORIZE THESE!
3 sides Triangle4 sides Quadrilateral5 sides Pentagon6 sides Hexagon7 sides Heptagon8 sides Octagon9 sides Nonagon10 sides Decagon11 sides Undecagon12 sides Dodecagonn sides n-gon
![Page 10: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/10.jpg)
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
![Page 11: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/11.jpg)
Example #1
1. Name________. Is it concave or convex?__________
2. Name ________ Is it concave or convex?__________
1 2
ABCDEF
concave
ABCDE
convex
CDEFAB
DEABC
![Page 12: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/12.jpg)
Example #2
• Find the interior angle sum.
a. 13-gon b. decagon
(n – 2) 180(13 – 2) 180
(11) 1801980˚
(n – 2) 180(10 – 2) 180
(8) 1801440˚
![Page 13: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/13.jpg)
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 angles2. Solve for n
![Page 14: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/14.jpg)
Example #3
How many sides does each regular polygon have if its interior angle sum is:
a. 2700 b. 1080
2700 = (n – 2) 1802700 = (n – 2)
18015 = n – 2
17 = n17-gon
1080 = (n – 2) 1801080 = (n – 2)
1806 = n – 2
8 = nOctagon
Sum is given
![Page 15: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/15.jpg)
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 angles2. Solve for ONE
![Page 16: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/16.jpg)
Example #5Find y
First calculate pentagon sumPentagon sum = 540˚
540 = 5 y540 = y
5108˚ = y
Sum is calculated
![Page 17: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/17.jpg)
Example #6Find x.
Hexagon sum = 720˚
one angle of an equiangular hexagon
SUM = 720 = 120˚ 6 6
x makes a linear pair with an interior angle
x = 180˚ – 120˚ = 60˚ x = 60˚
120˚
![Page 18: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/18.jpg)
Summary:SUM of the Interior Angles of a Polygon
S = (n – 2) 180
One Interior Angle of a REGULAR PolygonOne = (n – 2) 180 = SUM
n n
![Page 19: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/19.jpg)
Example #7
Find x.
Heptagon sum = 900°
900 = x + 816132100155142167
+120816
84 = x
![Page 20: Chapter 5 5.1 The Polygon Angle-Sum Theorem HOMEWORK: Lesson 5.1/1-14.](https://reader033.fdocuments.in/reader033/viewer/2022061507/56649f115503460f94c2419b/html5/thumbnails/20.jpg)