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**Sec. 3-5 The Polygon Angle-Sum Theorems**

Objectives: To classify Polygons To find the sums of the measures of the interior & exterior s of Polygons.

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**Polygon: A closed plane figure. w/ at least 3 sides (segments)**

The sides only intersect at their endpoints Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.

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**Which of the following figures are polygons?**

No yes No

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**Example 1: Name the 3 polygons**

Top XSTU S T Bottom WVUX X U Big STUVWX V W

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**I. Classify Polygons by the number of sides it has.**

3 4 5 6 7 8 9 10 12 n Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon N-gon Interior Sum

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**II. Also classify polygons by their Shape**

a) Convex Polygon – Has no diagonal w/ points outside the polygon. E A D B C b) Concave Polygon – Has at least one diagonal w/ points outside the polygon. * All polygons are convex unless stated otherwise.

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**III. Polygon Interior sum**

4 sides 2 Δs 2 • 180 = 360 5 sides 3 Δs 3 • 180 = 540

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**S = (n -2) 180 All interior sums are multiple of 180°**

6 sides 4 Δs 4 • 180 = 720 All interior sums are multiple of 180° Th(3-9) Polygon Angle – Sum Thm S = (n -2) 180 Sum of Interior # of sides

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**Examples 2 & 3: Find the interior sum of a 15 – gon. S = (n – 2)180**

Find the number of sides of a polygon if it has an sum of 900°. S = (n – 2)180 900 = (n – 2)180 5 = n – 2 n = 7 sides

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**Special Polygons: Equilateral Polygon – All sides are .**

Equiangular Polygon – All s are . Regular Polygon – Both Equilateral & Equiangular.

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**IV. Exterior s of a polygon.**

1 3 2 1 2 3 4 5

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**Th(3-10) Polygon Exterior -Sum Thm**

The sum of the measures of the exterior s of a polygon is 360°. Only one exterior per vertex. 1 2 3 m1 + m2 + m3 + m4 + m5 = 360 5 4 For Regular Polygons The interior & the exterior are Supplementary. = measure of one exterior Int + Ext = 180

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Example 4: How many sides does a polygon have if it has an exterior measure of 36°. = 36 360 = 36n 10 = n

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**Example 5: S = (n - 2)180 = (15 – 2)180 = 24 = (13)180 = 2340 n = 15**

Find the sum of the interior s of a polygon if it has one exterior measure of 24. S = (n - 2)180 = (15 – 2)180 = (13)180 = 2340 = 24 n = 15

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**Example 6: Solve for x in the following example. 4 sides**

Total sum of interior s = 360 100 x = 360 280 + x = 360 x = 80

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Example 7: Find the measure of one interior of a regular hexagon. S = (n – 2)180 = (6 – 2)180 = (4)180 = 720 = 120

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Problem: What is the degree measure of each interior angle and exterior angle in a regular 18-gon? 18-gon: polygon with 18 sides regular: all angles are.

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