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

Holt McDougal 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

Holt McDougal 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

Holt McDougal 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

Holt McDougal 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

Holt McDougal Geometry 6-1 Properties and Attributes 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.

Holt McDougal 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons A regular polygon is one that is both equilateral and equiangular.

Holt McDougal 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

Holt McDougal 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

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

Holt McDougal Geometry 6-1 Properties and Attributes 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember!

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Holt McDougal Geometry 6-1 Properties and Attributes 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°.

Holt McDougal 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.

Holt McDougal 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. (5 – 2)180° = 540° Polygon  Sum Thm. mA + mB + mC + mD + mE = 540° Polygon  Sum Thm. 35c + 18c + 32c + 32c + 18c = 540Substitute. 135c = 540Combine like terms. c = 4Divide both sides by 135.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 3a Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° (15 – 2)180° 2340° Polygon  Sum Thm. A 15-gon has 15 sides, so substitute 15 for n. Simplify.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. Step 2 Find the measure of one interior angle. Check It Out! Example 3b (n – 2)180° (10 – 2)180° = 1440° Polygon  Sum Thm. Substitute 10 for n and simplify. The int. s are , so divide by 10.

Holt McDougal 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°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Holt McDougal 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.