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5.1 The Polygon Angle-Sum Theorem

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 equiangular Note: A regular polygon is always convex

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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.

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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* 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

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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

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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

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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

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Example #1 1. Name________. 2. Name ________

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

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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˚

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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

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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

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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

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Example #5 Find y 108˚ = y First calculate pentagon sum

Sum is calculated 540 = 5 y 540 = y 5 108˚ = y

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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˚

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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

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Example #7 Find x. 84 = x 900 = x + 816 Heptagon sum = 900° 132 100

155 142 167 +120 816 84 = x

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