1. Day 1 Properties of polygons
Have paper, a pencil and calculator on your desk

2. A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

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.

4. Example 1: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
Triangle
Hexagon
Octagon
Pentagon
N-gon
Heptagon
Nonagon
Not a polygon
polygon, hexagon

5. Example 2: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, heptagon

6. Example 3: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
not a polygon

7. Check It Out! Example 4
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon

8. Check It Out! Example 5
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
polygon, nonagon

9. Check It Out! Example 6
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon

10. 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.

11. 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.

12. Example 7: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
Regular, Concave
Regular, Convex
Irregular, Concave
Irregular, Convex
irregular, convex

13. Example 8: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

14. Example 9: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

15. To find the sum of the interior angle measures 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.

17. The measure of each interior angle of a regular n-gon is :

18. Example 10: Finding Interior Angle Measures and Sums in Polygons
Find the sum of the interior angle measures of a
convex heptagon.
(n – 2)180°
Polygon  Sum Thm.
(7 – 2)180°
A heptagon has 7 sides, so substitute 7 for n.
900°
Simplify.

19. Example 11: Finding Interior Angle Measures and Sums in Polygons
Find the measure of each interior angle of a regular 16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
Substitute 16 for n and simplify.
(16 – 2)180° = 2520°
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.

20. Example 12: Finding Interior Angle Measures and Sums in Polygons
Find the Value of C
Polygon  Sum Thm.
(5 – 2)180° = 540°
Polygon  Sum Thm.
mA + mB + mC + mD + mE = 540°
35c + 18c + 32c + 32c + 18c = 540
Substitute.
135c = 540
Combine like terms.
c = 4
Divide both sides by 135.

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

22. In the polygon below, an exterior angle has been measured at each vertex. Notice the sum of the exterior angle measures is 360°.
For any polygon, the exterior angle and the interior angle at the same vertex must be a linear pair. So the interior angles measure to be:
180-41= 139°
180-55= 125°
=70°
180-43=137°
=69°

23. The measure of each exterior angle of a regular n-gon is:
360 n

24. Example 13: Finding Interior Angle Measures and Sums in Polygons
Find the measure of each exterior angle of a regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°.
Polygon  Sum Thm.
A regular 20-gon has 20  ext. s, so divide the sum by 20.
measure of one ext.  =
The measure of each exterior angle of a regular 20-gon is 18°.

25. Example 14: Finding Interior Angle Measures and Sums in Polygons
Find the value of b in polygon FGHJKL.
Polygon Ext.  Sum Thm.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
120b = 360
Combine like terms.
b = 3
Divide both sides by 120.

26. Lesson Quiz
1. Find the sum of the interior angle measures of a convex 11-gon.
2. Find the measure of each interior angle of a regular 18-gon.
3. Find the measure of each exterior angle of a regular 15-gon.

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