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

2. Warm Up 6.1 Properties of Polygons 1. A ? is a three-sided polygon.
2. A ? is a four-sided polygon.
Evaluate each expression for n = 6.
3. (n – 4) 12
4. (n – 3) 90
Solve for a.
5. 12a + 4a + 9a = 100
triangle
quadrilateral 24
270 4

3. Objectives Classify polygons based on their sides and angles.
6.1 Properties of Polygons
Objectives
Classify polygons based on their sides and angles.
Find and use the measures of interior and exterior angles of polygons.

4. Vocabulary side of a polygon vertex of a polygon diagonal
6.1 Properties of Polygons
Vocabulary
side of a polygon
vertex of a polygon
diagonal
regular polygon
concave
convex

5. 6.1 Properties of Polygons
In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons.

6. 6.1 Properties of Polygons
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.

7. 6.1 Properties of Polygons
You can name a polygon by the number of its sides. The table shows the names of some common polygons.

8. 6.1 Properties of Polygons
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!

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

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

11. Example 1C: Identifying Polygons
6.1 Properties of Polygons
Example 1C: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
not a polygon

12. 6.1 Properties of Polygons
Check It Out! Example 1a
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon

13. 6.1 Properties of Polygons
Check It Out! Example 1b
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
polygon, nonagon

14. 6.1 Properties of Polygons
Check It Out! Example 1c
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon

15. 6.1 Properties of Polygons
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.

16. 6.1 Properties of Polygons
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.

17. Example 2A: Classifying Polygons
6.1 Properties of Polygons
Example 2A: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex

18. Example 2B: Classifying Polygons
6.1 Properties of Polygons
Example 2B: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

19. Example 2C: Classifying Polygons
6.1 Properties of Polygons
Example 2C: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

20. 6.1 Properties of Polygons
Check It Out! Example 2a
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

21. 6.1 Properties of Polygons
Check It Out! Example 2b
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

22. 6.1 Properties of Polygons
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.

23. 6.1 Properties of Polygons
By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.
Remember!

24. 6.1 Properties of Polygons

25. 6.1 Properties of Polygons
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°.

26. Example 3A: Finding Interior Angle Measures and Sums in Polygons
6.1 Properties of Polygons
Example 3A: 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.

27. Example 3B: Finding Interior Angle Measures and Sums in Polygons
6.1 Properties of Polygons
Example 3B: 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.

28. Example 3C: Finding Interior Angle Measures and Sums in Polygons
6.1 Properties of Polygons
Example 3C: Finding Interior Angle Measures and Sums in Polygons
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.

29. 6.1 Properties of Polygons
Example 3C Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°

30. 6.1 Properties of Polygons
Check It Out! Example 3a
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.

31. 6.1 Properties of Polygons
Check It Out! Example 3b
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.

32. 6.1 Properties of Polygons
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°.

33. 6.1 Properties of Polygons
An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!

34. 6.1 Properties of Polygons

35. Example 4A: Finding Interior Angle Measures and Sums in Polygons
6.1 Properties of Polygons
Example 4A: 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°.

36. Example 4B: Finding Interior Angle Measures and Sums in Polygons
6.1 Properties of Polygons
Example 4B: 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.

37. 6.1 Properties of Polygons
Check It Out! Example 4a
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°.

38. 6.1 Properties of Polygons
Check It Out! Example 4b
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.

39. Example 5: Art Application
6.1 Properties of Polygons
Example 5: Art Application
Ann is making paper stars for party decorations. What is the measure of 1?
1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°.
A regular pentagon has 5  ext. , so divide the sum by 5.

40. 6.1 Properties of Polygons
Check It Out! Example 5
What if…? Suppose the shutter were formed by 8 blades instead of 10 blades. What would the measure of each exterior angle be?
CBD is an exterior angle of a regular octagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°.
A regular octagon has 8  ext. , so divide the sum by 8.

41. 6.1 Properties of Polygons
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.
nonagon; irregular; concave
1620°
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.
160°
24°