3 6 Angles in Polygons Date Sums of

3. 6 Angles in Polygons Date: _____

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

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

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

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

Interior Angles 2 4 3 1 5 6 Exterior Angles

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

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

Sum of Exterior Angles 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 180 180 180 • 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 3 540 180 360 Quadrilateral 4 720 360 Pentagon 5 900 540 360 Hexagon 6 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 ! Each Interior Angle 180(n – 2) n Each Exterior Angle 360 n

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

The sum of the interior angles of a convex polygon is 1440. How many sides does the polygon have? 180 • (n – 2) = 1440 180 n – 360 = 1440 + 360 +360 180 n = 1800 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. 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 exterior angle 360 =6 60

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.