1. 6.1: Properties of Polygons
Objectives
Classify polygons based on their sides and angles.

2. A segment that connects non-consecutive vertices.
Vocabulary
Sketch
Things to Remember
Polygon
A closed figure formed by 3 or more segments that intersect only at their endpoints.
Diagonal
A segment that connects non-consecutive vertices.
Quadrilateral
A four-sided polygon.
Pentagon
A five-sided polygon.
Hexagon
A six-sided polygon. 4 5 6

3. Heptagon Octagon Nonagon Decagon Dodecagon Vocabulary Sketch
Things to Remember
Heptagon
A seven-sided polygon.
Octagon
An eight-sided polygon.
Nonagon
A nine-sided polygon.
Decagon
A ten-sided polygon.
Dodecagon
A twelve-sided polygon. 7 8     9      10 12

4. n-gon Regular Polygon Irregular Polygon Concave Polygon Convex Polygon
Vocabulary
Sketch
Things to Remember
n-gon
A polygon with n number of sides.
Regular Polygon
An equilateral AND equiangular polygon.
Irregular Polygon
Any polygon that is NOT regular.
Concave Polygon
Any part of the diagonal lies in the exterior of the polygon.
Convex Polygon
All diagonals lie in the interior of the polygon.

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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.

13. 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.

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

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

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

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

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

19. 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.

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

22. 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°.

23. 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.

24. 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.

25. 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.

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

27. 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.

28. 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.

29. 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°.

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

32. 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°.

33. 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.

34. 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°.

35. 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.

36. 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.

37. 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.

38. 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°