Angles in Polygons.

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Presentation transcript:

Angles in Polygons

Sums of Interior Angles Triangle Quadrilateral Pentagon = 2 triangles = 3 triangles Hexagon Octagon = 4 triangles Heptagon = 5 triangles = 6 triangles

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

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

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

Interior Angles 1 2 3 4 5 6 Exterior Angles

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

Sums of Exterior Angles 180 180 180 180•4 = 720 180 Sum of Interior & Exterior Angles = 720 Sum of Interior Angles = 360 Sum of Exterior Angles = 720- 360= 360

Sum of Exterior Angles 180•5 = 900 Sum of Interior & Exterior Angles = 180 180 180 180 180 180•5 = 900 Sum of Interior & Exterior Angles = 900 Sum of Interior Angles = 540° Sum of Exterior Angles = 900- 540= 360

Sum of Exterior Angles 180•6 = 1080 180 180 180 180 180 180 180•6 = 1080 Sum of Interior & Exterior Angles = 1080 Sum of Interior Angles = 720° Sum of Exterior Angles = 1080- 720= 360

Sums of Exterior Angles Polygon # of Sides Interior + Exterior Interior Angles Exterior Angles Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 540 180 360 720 360 360 900 540 360 1080 720 360 Sum of Exterior Angles is always 360!

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

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

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

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

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

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 Find the measure of each exterior angle of a