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Sec. 3-5 The Polygon Angle-Sum Theorems Objectives: a) To classify Polygons b) To find the sums of the measures of the interior & exterior s of Polygons.

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Polygon: A closed plane figure. A closed plane figure. w/ at least 3 sides (segments) w/ at least 3 sides (segments) The sides only intersect at their endpoints 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. 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? yesNo

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Example 1: Name the 3 polygons S T U V W X Top XSTU Bottom WVUX Big STUVWX

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I. Classify Polygons by the number of sides it has. Sides nName 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. EA B C D 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 = sides 3 Δs = 540

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

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Examples 2 & 3: Find the interior sum of a 15 – gon. Find the interior sum of a 15 – gon. S = (n – 2)180 S = (15 – 2)180 S = (13)180 S = 2340 Find the number of sides of a polygon if it has an sum of 900°. Find the number of sides of a polygon if it has an sum of 900°. S = (n – 2) = (n – 2)180 5 = n – 2 n = 7 sides

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Special Polygons: Equilateral Polygon – All sides are. Equilateral Polygon – All sides are. Equiangular Polygon – All s are. Equiangular Polygon – All s are. Regular Polygon – Both Equilateral & Equiangular. Regular Polygon – Both Equilateral & Equiangular.

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IV. Exterior s of a polygon

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

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

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Example 5: Find the sum of the interior s of a polygon if it has one exterior measure of 24. Find the sum of the interior s of a polygon if it has one exterior measure of 24. = 24 n = 15 S = (n - 2)180 = (15 – 2)180 = (13)180 = 2340

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Example 6: Solve for x in the following example. Solve for x in the following example. x sides Total sum of interior s = x = x = 360 x = 80

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

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