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Angles in Polygons

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Sums of Interior Angles TriangleQuadrilateralPentagon Heptagon Octagon Hexagon = 2 triangles = 3 triangles = 4 triangles = 5 triangles= 6 triangles

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Convex Polygon # of Sides# of Triangles Sum of Interior Angles Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon 3 4 5 6 7 8 n 1 2 3 4 5 6 n – 2 180 360 540 720 900 1080 180(n – 2)

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Find the measure of the missing angle in the figure below 100 135 70 x 135 + 100 + 70 + x = quadrilateral 360 305 + x = 360 -305 x = 55

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m 1 = 1 2 3 110 (5x - 5) (4x + 15) (8x - 10) pentagon 5x - 5 + 4x + 15 + 8x - 10 + 110 + 90 =540 17x + 200= 540 -200 -200 17x = 340 x = 20 17 17 5(20) - 5 = 95 Find m 1.

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1 2 3 4 5 6 Exterior Angles Interior Angles

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Sum of Interior Angles = Sum of Interior & Exterior Angles = 180 1 2 3 4 5 6 540 Sum of Exterior Angles = 360 540 - 180 = Sums of Exterior Angles 1803 = 540

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180 Sum of Interior Angles = Sum of Interior & Exterior Angles = 360 720 Sum of Exterior Angles = 360 720 - 360 = Sums of Exterior Angles 1804 = 720

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Sum of Exterior Angles 180 Sum of Interior Angles = Sum of Interior & Exterior Angles = Sum of Exterior Angles = 360 900 - 540 = 900 540° 1805 = 900

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Sum of Exterior Angles Sum of Interior Angles = Sum of Interior & Exterior Angles = Sum of Exterior Angles = 180 360 1080 - 720 = 1080 720° 1806 = 1080

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Sums of Exterior Angles Polygon# of Sides Interior + Exterior Interior Angles Exterior Angles Triangle3 Quadrilateral4 Pentagon5 Hexagon6 180 360 540 720 540 720 900 1080 360 Sum of Exterior Angles is always 360 !

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Angles of Regular Polygons Sum of the Interior Angles Sum of the Exterior Angles Each Interior Angle n Each Exterior Angle n 180(n – 2) Always 360 ! 180(n – 2) 360

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Find the sum of the measures of the interior angles of a regular dodecagon. 180(n – 2) = 180(12 – 2) n = 12 = 180(10) = 1800 all 12 angles = 150 1800 12 each angle What is the measure of each angle?

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The sum of the interior angles of a convex polygon is 1440. How many sides does the polygon have? 180(n – 2) = 1440 180n = 1800 + 360 +360 180 n = 10 10 sides 180n – 360 = 1440

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Exterior Angles What is the measure of each exterior angle of a regular hexagon? 360 6 sides 6 = 60

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The measure of each exterior angle of a regular polygon is 20. How many sides does it have? 20 360 =18 The measure of each interior angle of a regular polygon is 120. How many sides does it have? 60 360 =6 180 - 120 = 60 exterior angle

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Find the sum of the interior angles of a 100-gon! Find the sum of the exterior angles of a 100-gon. Find the measure of each interior angle of a 100-gon. Find the measure of each exterior angle of a 100-gon.

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