Objectives:Develop and use formulas for the sums of the measures of interior and exterior angles of polygons

3.6 Angles in Polygons

Warm-Up: Here’s a two part puzzle designed to prove that half of eleven is six. First rearrange two sticks to reveal the number eleven. Then remove half of the sticks to reveal the number six.

Convex Polygon:

A polygon in which any line segment connecting two points of the polygon has no part outside the polygon.

Concave Polygon:

A polygon that is not convex.

Consider the following Pentagon:

𝟏

Divide the polygon into three triangular regions by drawing all the possible diagonals from one vertex.

Find each of the following:

𝟐

𝟑𝟒

𝟔

𝟓𝟗

𝟕𝟖

Add the three expressions:

Note: You can form triangular regions by drawing all possible diagonals from a given vertex of any polygon

Polygon # of sides

# of triangular

regions

Sum of Interiorangles

Triangle

Quadrilateral

Pentagon

Hexagon

3

4

5

n-gon

6

n

1

2

3

4

n2

180

360

540

720

180(n-2)

The sum of the measures of the interior angles of a polygon with n sides is:

180(n-2)

Note: Recall that a regular polygon is on in which all the angles are congruent.

Polygon # of sides

Measure of Interior

angles

Sum of Interiorangles

Triangle

Quadrilateral

Pentagon

Hexagon

3

4

5

n-gon

6

n

180

360

540

720

180(n-2)

60

90

108

120180(n-2)

n

The measure of an Interior Angle of a Regular Polygon with n sides is:

180(n-2) n

Exterior Angle Sums in Polygons

Polygon # of sides

Sum of interior &exterior angles

Sum of Interiorangles

Triangle

Quadrilateral

Pentagon

Hexagon

3

4

5

n-gon

6

n 180n

180

720

540

720

180(n-2)

Sum of Exteriorangles

360

360

360

360

360540

360

900

1080

Theorem

𝟑𝟔𝟎𝟎

Sum of the measures of the Exterior Angles of a Polygon is:

For a Convex Polygon

For a Regular Polygon

 Polygon

Numberof

Sides

Number of

ΔRegio

ns

Sum of Interior Angles

Sum of

Exterior

Angles

Sum of Int & Ext Angle

s

Measure of

Interior Angles

Measure of

Exterior Angles

Triangle 3 1 180 360 540 60 120

Quadrilateral

4 2 360 360 720 90 90

Pentagon 5 3 540 360 900 108 72

Hexagon 6 4 720 360 1080

120 60

Heptagon 7 5 900 360 1260

128.6 51.4

Octagon 8 6 1080 360 1440

135 45

Nonagon 9 7 1260 360 1620

140 40

Decagon 10 8 1440 360 1800

144 36

11-gon 11 9 1620 360 1980

147.3 32.7

Dodecagon 12 10 1800 360 2160

150 30

13-gon 13 11 1980 360 2340

152.3 27.2

n-gon n n-2 180(n-2)

360 180n

Find the indicated angle measures.

𝟓𝟒𝟎

𝟒𝟖𝟎

𝒙𝟎

𝟏𝟎𝟎𝟎

𝟏𝟏𝟓𝟎

𝟏𝟐𝟎𝟎

𝒛𝟎

𝒚𝟎

Find the indicated angle measures.

𝟗𝟎𝟎

𝒙𝟎

𝟕𝟓𝟎

𝟖𝟓𝟎 𝟏𝟑𝟐𝟎

𝟏𝟐𝟎𝟎

𝒚𝟎

Find the indicated angle measures.

𝟗𝟎𝟎

𝒙𝟎 𝟏𝟎𝟎𝟎

𝟏𝟑𝟎𝟎

𝟏𝟏𝟎𝟎

Find the indicated angle measures.

𝟏𝟎𝟓𝟎𝒙𝟎

𝟏𝟎𝟎𝟎

𝟏𝟏𝟎𝟎

𝟕𝟓𝟎

For each polygon determine the measure of an interior angle and the measure of an exterior angle.

A rectangle

A regulardodecagon

An equilateraltriangle

An equiangularpentagon

An interior angle measure of a regular polygon is given. Find the number of sides of the polygon

𝟏𝟑𝟓𝟎 𝟏𝟓𝟎𝟎 𝟏𝟔𝟓𝟎

An exterior angle measure of a regular polygon is given. Find the number of sides of the polygon

𝟔𝟎𝟎 𝟑𝟔𝟎 𝟐𝟒𝟎

Find the indicated angle measure.(𝟒 𝒙 )𝟎 (𝟑 𝒙)𝟎

(𝟐 𝒙)𝟎(𝒙)𝟎

Find the indicated angle measure.¿¿ (𝟐 𝒙+𝟑𝟎)𝟎

(𝟖 𝒙−𝟏𝟎)𝟎

(𝒙 ¿¿𝟐+𝟏𝟎)𝟎¿

Find the indicated angle measure.

¿¿

¿

¿¿