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Properties and Attributes of Polygons

Learn about the properties and attributes of polygons, including how to identify polygons based on their sides, classify polygons as regular or irregular, determine if they are concave or convex, find the sum of interior angle measures, and calculate the measure of each interior angle in different polygons.

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Properties and Attributes of Polygons

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

  2. You can name a polygon by the number of its sides. The table shows the names of some common polygons.

  3. 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

  4. 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

  5. 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

  6. 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

  7. All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

  8. 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.

  9. A regular polygon is one that is both equilateral and equiangular.

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

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

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

  13. 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.

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

  15. 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°.

  16. 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.

  17. 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.

  18. 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.

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

  20. 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.

  21. 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.

  22. 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°.

  23. 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.

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