Angles in Polygons

Sums of Interior Angles

Triangle Quadrilateral Pentagon

Heptagon OctagonHexagon

= 2 triangles = 3 triangles

= 4 triangles= 5 triangles = 6 triangles

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

180360

540

720

900

1080

180•(n – 2)

Find the measure of the missing angle in the figure below

100135

70x

135 + 100 + 70 + x =

quadrilateral

360305 + x = 360

-305 -305x = 55

m1 =1

2

3

110

(5x - 5)

(4x + 15)

(8x - 10)

pentagon5x - 5 + 4x + 15 + 8x - 10 + 110 + 90 = 540

17x + 200= 540 -200 -200

17x = 340

x = 20 17 17

5(20) - 5

= 95

Find m1.

12

34

5 6

Exterior Angles

Interior Angles

Sum of Interior Angles =

Sum of Interior & Exterior Angles =

180

12

34

5 6

180

180

180

540

Sum of Exterior Angles = 360 540- 180=

Sums of Exterior Angles

180•3 = 540

180

180

180

180

Sum of Interior Angles =

Sum of Interior & Exterior Angles =

360 720

Sum of Exterior Angles = 360 720- 360=

Sums of Exterior Angles

180•4 = 720

Sum of Exterior Angles

180 180

180

180

180

Sum of Interior Angles =

Sum of Interior & Exterior Angles =

Sum of Exterior Angles = 360 900- 540=

900

540°

180•5 = 900

Sum of Exterior Angles

Sum of Interior Angles = Sum of Interior & Exterior Angles =

Sum of Exterior Angles =

180

180180

180

180

180

360 1080- 720=

1080720°

180•6 = 1080

Sums of Exterior Angles

Polygon # of Sides

Interior +

Exterior

Interior Angles

Exterior Angles

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

180

360

540

720

540

720

900

1080

360

360

360

360Sum of Exterior Angles

is always 360!

Angles of Regular Polygons

Sum of the Interior Angles

Sum of the Exterior Angles

Each Interior Anglen

Each Exterior Angle n

180(n – 2)

Always 360!

180(n – 2)

360

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

= 150180012

each angle

What is the measure of 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

180n = 1800+ 360 +360

180 180

n = 10 10 sides

180n – 360 = 1440

Exterior Angles

What is the measure of each exterior angle of a regular hexagon?

360

6 sides

6= 60

The measure of each exterior angle of a regular polygon is 20. How many

sides does it have?

20360 = 18

The measure of each interior angle of a regular polygon is 120. How many

sides does it have?

60360 = 6

180 - 120 = 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 100-gon.

Find the measure of each exterior angle of a 100-gon.