Chapter 5

5.1 The Polygon Angle-Sum Theorem

HOMEWORK:Lesson 5.1/1-14

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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

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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

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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

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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

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

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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

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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

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

Naming A Polygon

A polygon is named by the number of_____.

ex: If a polygon has

___ sides, you use

___ letters.

Polygon ABCDE

SIDES

5

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Example #1

1. Name________. Is it concave or convex?__________

2. Name ________ Is it concave or convex?__________

1 2

ABCDEF

concave

ABCDE

convex

CDEFAB

DEABC

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˚

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

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

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

Example #5Find y

First calculate pentagon sumPentagon sum = 540˚

540 = 5 y540 = y

5108˚ = y

Sum is calculated

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˚

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

Example #7

Find x.

Heptagon sum = 900°

900 = x + 816132100155142167

+120816

84 = x