1. 6-1 Properties and Attributes of Polygons Warm Up Lesson Presentation
Lesson Quiz
Holt Geometry

2. Warm Up 1. A _____ is a three-sided polygon.
2. A _____ ___ is a four-sided polygon.
Triangle
quadrilateral

3. Objectives Classify polygons based on their sides and angles.
Find and use the measures of interior and exterior angles of polygons.
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!

4. side of
Each segment that forms a polygon is a ________ _____________.
The common endpoint of two sides is a ________ _________ .
A segment that connects any two nonconsecutive vertices is a __________.
the polygon
vertex of
the polygon
diagonal

5. 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.

6. Example 1A: Identifying Polygons
1. Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, hexagon

7. Example 1B: Identifying Polygons
2. Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, heptagon

8. Example 1C: Identifying Polygons
3. Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
not a polygon because the figure has a curved side

9. Check It Out! Example 1a
4. Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon because it’s not a closed figrue

10. Check It Out! Example 1b
5. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
polygon, nonagon

11. Check It Out! Example 1c
6. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon because it has a curved side

12. 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.

13. Example 2A: Classifying Polygons
7. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex

14. Example 2C: Classifying Polygons
8. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

15. Example 2B: Classifying Polygons
9. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

16. Check It Out! Example 2a
10. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

17. Check It Out! Example 2b
11. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

18. 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.

19. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.
Remember!

20. In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles
is (n — 2)180°.

21. Example 3A: Finding Interior Angle Measures and Sums in Polygons
12. 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.

22. Example 3B: Finding Interior Angle Measures and Sums in Polygons
13. 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.

23. Example 3C: Finding Interior Angle Measures and Sums in Polygons
14. Find the measure of each interior angle of pentagon ABCDE.
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.

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

25. Check It Out! Example 3a
15. Find the sum of the interior angle measures of a convex 15-gon.
(n – 2)180°
Polygon  Sum Thm.
(15 – 2)180°
A 15-gon has 15 sides, so substitute 15 for n.
2340°
Simplify.

26. Check It Out! Example 3b
16. Find the measure of each interior angle of a regular decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
Substitute 10 for n and simplify.
(10 – 2)180° = 1440°
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 10.

27. 17. 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°.

28. An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!

30. Example 4A: Finding Interior Angle Measures and Sums in Polygons
18. 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°.

31. Example 4B: Finding Interior Angle Measures and Sums in Polygons
19. 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.

32. Check It Out! Example 4a
20. Find the measure of each exterior angle of a regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
Polygon  Sum Thm.
A regular dodecagon has 12  ext. s, so divide the sum by 12.
measure of one ext.
The measure of each exterior angle of a regular dodecagon is 30°.

33. Check It Out! Example 4b
21. Find the value of r in polygon JKLM.
4r° + 7r° + 5r° + 8r° = 360°
Polygon Ext.  Sum Thm.
24r = 360
Combine like terms.
r = 15
Divide both sides by 24.