1. A polygon is a closed figure whose sides are line segments that intersect only at their endpoints.
In a regular polygon , all the angles have the same measures and all the sides have the same length.
Polygons can be indentified by the number of their sides
Regular Polygons
Not Polygons
Polygons

2. Practice- Is the shape a polygon, a regular polygon or not a polygon?1. 2. 3.
Not a Polygon –
The sides are not line segments.
Regular Polygon – All the sides and angles have the same measure.
Polygon – All the sides are line segments.

4. A vertex is a point where two sides meet.
The interior angles of a polygon are the angles inside the polygon.
A diagonal of a polygon is a line segment that joins two nonconsecutive vertices.
Vertex
What is the sum of the measures of the interior angles of a quadrilateral?
Interior Angles
Since the quadrilateral is made up of 2 triangles, the sum the angles is 2(180o) = 360o
Diagonal

5. Sum of angle measures in an n-gon = (n-2) 180
You can use triangles to find these sum of the angle measures in other figures.
Shape
Quadrilateral
Pentagon
Hexagon
Heptagon
Number of Sides
Number of Diagonals from one vertex
Number of Triangles Formed
Sum of Angle Measures
13-gon
n-gon n
n-2
(n-2)180o 4 1 2
2(180o)=360o 5 2 3
3(180o)=540o 6 3 4
4(180o)=720o 7 4 5
5(180o)=900o 13 10 11
11(180o)
=1980o
Sum of angle measures in an n-gon = (n-2) 180

6. Sum of an angle measures in an n-gon = (n-2) 180
1. Find the sum of the measures of the interior angles of octagon.
An octagon has 8 sides, so n=8.
(n-2)180
(8-2)180
(6)180
1,080 o
2. Find the sum of the measures of the interior angles of an 11-gon.
A 11-gon has 11 sides, so n=11.
(n-2)180
(11-2)180
(9)180
1,620 o

7. Find the sum of the measures of the interior angles in the polygon.
3) 14-gon ) 17-gon ) 19-gon

8. Remember, in a regular polygon, all the angles have the same measure.
Hexagon
120 o
Hexagon
120 o
120 o
6) Find the measure of one angle in a regular hexagon.
(n-2)180
(6-2)180
(4)180
720 o
720 ÷ 6 = 120 o

9. Sum of an angle measures in an n-gon = (n-2) 180
Measure of one angle in a regular n-gon = (n-2) 180. n
8) Find the measure of one angle in a regular 14-gon. Round to the nearest tenth of a degree.
7) Find the measure of one angle in a regular octagon.
A regular octagon has 8 sides, so n=8.
A regular 14-gon has 14 sides, so n=14.
= (n-2)180 n
= (n-2)180 n
= (8-2)180 8
= (14-2)180 14
= (6)180 8
= (12)180 14
= 1080 8
= 135 o
= 2160 14
= o
One angle in a regular octagon is 135 o.
One angle in a regular 14-gon is o.

10. Find the measure of one angle in the polygon.
9) Regular 12-gon ) Regular 16-gon ) Regular 20-gon

11. Sum of an angle measures in an n-gon = (n-2) 180
Indentify the Polygon
12) The sum of the measures of the interior angles in a polygon is 1,440 o. What type of polygon is it?
Sum of an angle measures in an n-gon = (n-2) 180
1440 = (n-2)180
1440 = 180n-360
1800 = 180n
10 = n
A polygon interior angles that has a sum of 1440o is a decagon.

12. Practice- Identify the polygon whose interior angles have the following sum.
13) o ) 1260o ) 3420o

13. Find the value of x. 17) 16) 98o xo 140o 175o 150o xo 115o (n-2)180
(6-2)180
720 = 563 +x
(4)180
157 = x
=720 o

14. Find the value of x.
18) )

15. 20)

16. 20) ) )

17. 23) ) )