1. Lesson 6 – 1 Angles of Polygons
Geometry
Lesson 6 – 1
Angles of Polygons
Objective:
Find and use the sum of the measures of the interior angles of a polygon.
Find and use the sum of the measures of the exterior angles of a polygon.

2. Diagonals
A diagonal of a polygon is a segment that connects any two nonconsecutive vertices.

3. How many diagonals can be drawn from 1 vertex of a convex polygon?
Quadrilateral
Triangle
Pentagon
Hexagon

4. 3 1
1(180) = 180 4 2
2(180) = 360 5 3
3(180) = 540
4(180) = 720 6 4
n-gon n
n - 2
(n-2)(180)

5. Theorem Polygon Interior Angles Sum
The sum of the interior angle measures of an n-sided convex polygon is (n-2)(180).

6. Find the sum of the interior angles of a convex heptagon.
Find the sum of the measures of the interior angles of a convex octagon.
(n-2)(180)
(7-2)(180)
(5)(180)
900
(8-2)(180)
6(180)
1080

7. Find the measure of each interior angle of quadrilateral ABCD.
3x + x = 360
(n-2)(180)
2(180)
360
4x = 360
4x = 180
x = 45

8. Find the measure of each interior angle of pentagon HJKLM.
(5-2)(180)
3(180)
540
2x x + 3x x + 14 = 540
10x = 540
10x = 370
x = 37

9. Sum of the interior angles: 720 Since it is a regular hexagon, all
The poles for a tent form the vertices of a regular hexagon. When the poles are properly positioned, what is the measure of the angle formed at a corner of the tent?
(n – 2)(180)
(6 – 2)(180)
4(180)
720
Sum of the interior angles: 720
Since it is a regular hexagon, all
sides and angles are equal.
To find one angle take the total sum divided by the number of sides.
One angle: 720/6 = 120

10. Find the measure of each interior angle of the regular 11-gon that appears on the face of a Susan B. Anthony one-dollar coin.
One angle:
(11-2)(180)
9(180)
1620
Don’t round unless they tell you to!

11. Find the number of sides.
The sum of the measures of the interior angles of a regular polygon is given. Find the number of sides in the polygon
Sum of 720
What is the formula for sum of interior angles?
(n – 2)(180) = sum of interior angles
(n – 2)(180) = 720
180n – 360 = 720
180n = 1080
n = 6 sides

12. Find the number of sides.
The sum of the measures of the interior angles of a regular polygon is given. Find the number of sides in the polygon
Sum of 1800
What is the formula for sum of interior angles?
(n – 2)(180) = sum of interior angles
(n – 2)(180) = 1800
180n – 360 = 1800
180n = 2160
n = 12 sides

13. Find the number of sides
The measure of an interior angle of a regular polygon is Find the number of sides in the polygon.
(n – 2)(180) = sum of interior angles
(n – 2)(180) = 135n
To get the sum take 135
times the number of sides n
180n – 360 = 135n
45n = 360
n = 8 sides

14. Find the number of sides
The measure of an interior angle of a regular polygon is Find the number of sides in the polygon.
(n – 2)(180) = sum of interior angles
(n – 2)(180) = 144n
To get the sum take 144
times the number of sides n
180n – 360 = 144n
36n = 360
n = 10 sides

15. Exterior Angles
Look at the following figures. Each vertex has one exterior angle identified (linear pair with interior angle).
What do the three figures have in common?
=360
= 360
=360

16. Theorem Polygon Exterior Angles Sum
The sum of the exterior angle measures of a convex polygon, one at each vertex, is 360.

17. Find the value of x in the diagram
3x x – 5 + 5x + 2x + 6x – 5 = 360
18x = 360
x = 20

18. Find the measure of each exterior angle of a regular nonagon.
One exterior angle =
sum of exterior angles / 1 exterior angle
One exterior angle = 360 / 9
= 40

19. Find the value of x in the diagram.
Find the measure of each exterior angle of a regular dodecagon.
x + 9x + 2x = 360
17x = 360
17x = 221
x = 13
One exterior angle = 360 / 12
= 30

20. Homework Pg. 393 1 – 11 all, 12 – 40 EOE, 50, 54 – 66 E
* Only write congruence statement on 62 & 64