1. 3.6 Angles in Polygons Objectives: Warm-Up:
Develop and use formulas for the sums of the
measures of interior and exterior angles of 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.

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

3. Concave Polygon:
A polygon that is not convex.

4. Consider the following Pentagon:𝟐 𝟑 𝟒 𝟏
Divide the polygon into three triangular regions by drawing all the possible diagonals from one vertex. 𝟓 𝟗 𝟔 𝟕 𝟖
Add the three expressions:
Find each of the following:
𝒎<𝟏+𝒎<𝟐+𝒎<𝟑= ______
𝒎<𝟒+𝒎<𝟓+𝒎<𝟔= ______
𝒎<𝟕+𝒎<𝟖+𝒎<𝟗= ______

5. Note: Polygon Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540
You can form triangular regions by drawing all possible diagonals from a given vertex of any polygon
# of
triangular
regions
Sum of
Interior
angles
Polygon
# of
sides
Triangle 3 1
180
Quadrilateral 4 2
360
Pentagon 5 3
540
Hexagon 6 4
720
n-gon n
n−2
180(n-2)

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

7. Note: Polygon Triangle 3 180 60 Quadrilateral 4 360 90 Pentagon 5 540
Recall that a regular polygon is on in which all the angles are congruent.
Sum of
Interior
angles
Measure
of Interior
angles
Polygon
# of
sides
Triangle 3
180 60
Quadrilateral 4
360 90
Pentagon 5
540
108
Hexagon 6
720
120
n-gon n
180(n-2) n
180(n-2)

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

9. Exterior Angle Sums in Polygons

10. Polygon Triangle 3 540 180 360 Quadrilateral 4 720 360 360 Pentagon 5
Sum of
interior &
exterior
angles
Sum of
Interior
angles
Sum of
Exterior
angles
Polygon
# of
sides
Triangle 3
540
180
360
Quadrilateral 4
720
360
360
Pentagon 5
900
540
360
Hexagon 6
1080
720
360
n-gon n
180n
360
180(n-2)

11. Theorem
Sum of the measures of the Exterior Angles of a Polygon is:
𝟑𝟔𝟎 𝟎

12. For a Convex Polygon
For a Regular Polygon
Polygon
Number of
Sides
Number of Δ
Regions
Sum of Interior Angles
Sum of Exterior Angles
Sum of Int & Ext Angles
Measure of
Interior Angles
Measure
Exterior Angles
Triangle 3 1
180
360
540 60
120
Quadrilateral 4 2
720 90
Pentagon 5
900
108 72
Hexagon 6
1080
Heptagon 7
1260
128.6
51.4
Octagon 8
1440
135 45
Nonagon 9
1620
140 40
Decagon 10
1800
144 36
11-gon 11
1980
147.3
32.7
Dodecagon 12
2160
150 30
13-gon 13
2340
152.3
27.2
n-gon n
n-2
180(n-2)
180n
𝟏𝟖𝟎(𝐧−𝟐) 𝐧
𝟏𝟖𝟎− 𝟏𝟖𝟎(𝐧−𝟐) 𝐧

13. Find the indicated angle measures.
𝟏𝟏𝟓 𝟎
𝟏𝟐𝟎 𝟎
𝟓𝟒 𝟎
𝒚 𝟎
𝒙 𝟎
𝒛 𝟎
𝟒𝟖 𝟎
𝟏𝟎𝟎 𝟎

14. Find the indicated angle measures.
𝒚 𝟎
𝟗𝟎 𝟎
𝟕𝟓 𝟎
𝒙 𝟎
𝟖𝟓 𝟎
𝟏𝟑𝟐 𝟎
𝟏𝟐𝟎 𝟎

15. Find the indicated angle measures.
𝟏𝟑𝟎 𝟎
𝒙 𝟎
𝟏𝟎𝟎 𝟎
𝟗𝟎 𝟎
𝟏𝟏𝟎 𝟎

16. Find the indicated angle measures.
𝒙 𝟎
𝟏𝟎𝟓 𝟎
𝟏𝟏𝟎 𝟎
𝟕𝟓 𝟎
𝟏𝟎𝟎 𝟎

17. A rectangle An equilateral triangle A regular dodecagon An equiangular
For each polygon determine the measure of an interior angle and the measure of an exterior angle.
A rectangle
An equilateral
triangle
A regular
dodecagon
An equiangular
pentagon

18. An interior angle measure of a regular polygon is given
An interior angle measure of a regular polygon is given. Find the number of sides of the polygon
𝟏𝟑𝟓 𝟎
𝟏𝟓𝟎 𝟎
𝟏𝟔𝟓 𝟎

19. An exterior angle measure of a regular polygon is given
An exterior angle measure of a regular polygon is given. Find the number of sides of the polygon
𝟔𝟎 𝟎
𝟑𝟔 𝟎
𝟐𝟒 𝟎

20. Find the indicated angle measure.
(𝟒𝒙) 𝟎
(𝟑𝒙) 𝟎
(𝒙) 𝟎
(𝟐𝒙) 𝟎

21. Find the indicated angle measure.
(𝒙 𝟐 + 𝟐𝒙+𝟏𝟎) 𝟎
(𝟐𝒙+𝟑𝟎) 𝟎
(𝟖𝒙−𝟏𝟎) 𝟎
(𝒙 𝟐 +𝟏𝟎) 𝟎

22. Find the indicated angle measure.
( 𝟓𝒙−𝟏𝟒) 𝟎
( 𝟒𝒙+𝟑𝟐) 𝟎
( 𝟒𝒙−𝟑𝟕) 𝟎
( 𝟐𝒙+𝟒) 𝟎
( 𝟓𝒙+𝟏𝟓) 𝟎