6 1 Properties and Attributes of Polygons Warm

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

6 -1 Properties and Attributes of Polygons Learning Targets I will classify polygons based on their sides and angles. I will find and use the measures of interior and exterior angles of polygons. Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Holt Mc. Dougal Geometry

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. Holt Mc. Dougal 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 Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Example 1 A: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon Holt Mc. Dougal Geometry polygon, heptagon not a polygon

6 -1 Properties and Attributes of Polygons Check It Out! Example 1 a Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons A regular polygon is a polygon that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. Holt Mc. Dougal 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. The easy way to remember: If it caves in, it is concave. If it does not cave in, it is convex. Holt Mc. Dougal Geometry

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

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

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

6 -1 Properties and Attributes of Polygons Holt Mc. Dougal 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 Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Example 3 A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of the following convex polygons. 1. Heptagon 1. 900° ° Holt Mc. Dougal Geometry 2. Decagon 2. 1, 440° 3. Pentagon 3. 540

6 -1 Properties and Attributes of Polygons Example 3 B: 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. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Example 3 C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. c=4 m A = 35(4°) = 140° m B = m E = 18(4°) = 72° m C = m D = 32(4°) = 128° Holt Mc. Dougal 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 Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Example 4 A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20 -gon. The measure of each exterior angle of a regular 20 -gon is 18°. Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons Check It Out! Example 4 b Find the value of r in polygon JKLM. 4 r° + 7 r° + 5 r° + 8 r° = 360° Polygon Ext. Sum Thm. 24 r = 360 r = 15 Combine like terms. Divide both sides by 24. Holt Mc. Dougal Geometry

6 -1 Properties and Attributes of Polygons HOMEWORK: Pg 398, #16 - 42 Holt Mc. Dougal Geometry