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 Warm Up 1. A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Evaluate each expression for n = (n – 4) (n – 3) 90 Solve for a a + 4a + 9a = 100 triangle quadrilateral

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Objectives

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

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons. A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Remember!

Holt McDougal 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 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 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

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons 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

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 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 Example 2C: Classifying Polygons 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 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 Check It Out! Example 2b 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 *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 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.  =

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° Polygon Ext.  Sum Thm. 120b = 360Combine like terms. b = 3Divide both sides by 120.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Find the measure of each exterior angle of a regular dodecagon. Check It Out! Example 4a A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. A regular dodecagon has 12  ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°. Polygon  Sum Thm. measure of one ext.

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.

Holt McDougal 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Lesson Quiz 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. nonagon; irregular; concave 2. Find the sum of the interior angle measures of a convex 11-gon. 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°

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Section 6.1 p. 398 #’s 2, 3, 7-9, 13, 15, 22, 26, 29, 31, 33, 37, 39, 45