1. Vocabulary side of a polygon vertex of a polygon diagonal
regular polygon
concave/convex
interior/exterior angles (i , e)

2. Each segment that forms a polygon is a side of the polygon
Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.
Where are interior/exterior angles? (i , e)

3. You can name a polygon by the number of its sides
You can name a polygon by the number of its sides. The table shows the names of some common polygons.
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!

4. Example 1: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, hexagon
polygon, heptagon
not a polygon

5. All the sides are congruent in an equilateral polygon
All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.
A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

6. Example 2: Classifying Polygons
irregular, convex
regular, convex
irregular, concave
regular, convex

7. To find the sum of the interior angle measures (Si) of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon.
Notice, there are (n – 2) triangles.

8. Convex Polygons
Si = (n-2) 180

9. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.
Se = 360

10. REGULAR POLYGONS REGULAR: me = 360/n OR n = 360/me
Also, you can use Si = n(i) for Regular
For REGULAR polygons always work with n, i , e
Remember, for any convex polygon:
mi+me = 180, Si = (n-2) 180, Se = 360

11. Example 3: Finding Interior Angle Measures and Sums in Polygons
Find the measure of each interior angle of pentagon ABCDE.
Si = (n-2) n = 5
Polygon  Sum Thm.
(5 – 2)180° = 540°
35c + 18c + 32c + 32c + 18c = 540
Substitute.
135c = 540
Combine like terms.
c = 4
Divide both sides by 135.

12. Example 3 Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°

13. Example 3B: Finding Interior Angle Measures and Sums in Polygons
Find the measure of each interior angle of a regular 18-gon. Then, find the sum of the angles.
Regular:
360/n = me → me= 20
mi + me = 180 → mi = 160
Si = n(i) → Si = 2880

14. Check It Out!
Find the measure of each interior angle of a regular decagon.
me = 36, mi = 144
If an interior angle of a regular polygon is
135, how many sides does it have?
me = 45, n = 8
If the sum of the interior angles is 2880,
how many sides does it have?
2880 = (n-2)180, n = 18
4. Can a polygon have Si = 3080?
3080 = (n-2)180, n = , NO!

15. READ CAREFULLY:
An interior or each interior versus sum of interior….
An exterior or each exterior versus sum of the exterior…
Regular or irregular…

16. Lesson Quiz
1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex.
2. Find the sum of the interior angle measures of a convex 11-gon.
3. Find the measure of each interior angle of a regular 18-gon.
4. Find the measure of each exterior angle of a regular 15-gon.