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

2. Holt Geometry 6-1 Properties and Attributes of Polygons 6-1 Properties and Attributes of Polygons Holt Geometry

3. 6-1 Properties and Attributes 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.

4. Holt Geometry 6-1 Properties and Attributes of Polygons You can name a polygon by the number of its sides. The table shows the names of some common polygons.

5. Holt Geometry 6-1 Properties and Attributes 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

6. Holt Geometry 6-1 Properties and Attributes 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

7. Holt Geometry 6-1 Properties and Attributes 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

8. Holt Geometry 6-1 Properties and Attributes 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

9. Holt Geometry 6-1 Properties and Attributes of Polygons A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

10. Holt Geometry 6-1 Properties and Attributes 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.

11. Holt Geometry 6-1 Properties and Attributes of Polygons Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex

12. Holt Geometry 6-1 Properties and Attributes of Polygons Example 2B: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

13. Holt Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 2b Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

14. Holt Geometry 6-1 Properties and Attributes of Polygons

15. Holt Geometry 6-1 Properties and Attributes 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° (7 – 2)180° 900° Polygon  Sum Thm. A heptagon has 7 sides, so substitute 7 for n. Simplify.

16. Holt Geometry 6-1 Properties and Attributes 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. Step 2 Find the measure of one interior angle. (n – 2)180° (16 – 2)180° = 2520° Polygon  Sum Thm. Substitute 16 for n and simplify. The int. s are , so divide by 16.

17. Holt Geometry 6-1 Properties and Attributes 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°.

18. Holt Geometry 6-1 Properties and Attributes of Polygons

19. Holt Geometry 6-1 Properties and Attributes 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°. A regular 20-gon has 20  ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°. Polygon  Sum Thm. measure of one ext.  =

20. Holt Geometry 6-1 Properties and Attributes 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 = 360Combine like terms. r = 15Divide both sides by 24.

21. Holt Geometry 6-1 Properties and Attributes 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.

22. Holt Geometry 6-1 Properties and Attributes of Polygons 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. Lesson Quiz 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°

23. Holt Geometry 6-1 Properties and Attributes of Polygons 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. Warm-Up(Pass back Papers!!!!!!!!!!!!!!!) 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°