11-1 Angle Measures in Polygons Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Warm Up 11.1 Angle Measures in Polygons 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

11.1 Angle Measures in Polygons Objectives Find and use the measures of interior and exterior angles of polygons.

Vocabulary side of a polygon vertex of a polygon diagonal 11.1 Angle Measures in Polygons Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex

11.1 Angle Measures in Polygons You learned that the name of a polygon depends on the number of sides. Now you will learn about the parts of a polygon and about ways to classify polygons.

11.1 Angle Measures in 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.

11.1 Angle Measures in Polygons Remember: You can name a polygon by the number of its sides. The table shows the names of some common polygons.

11.1 Angle Measures in Polygons A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Remember!

11.1 Angle Measures in 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.

11.1 Angle Measures in Polygons By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember!

6.1 Properties of Polygons

6.1 Properties 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°.  

Example 3A: Finding Interior Angle Measures and Sums in Polygons 6.1 Properties 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° Polygon  Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.

Example 3B: Finding Interior Angle Measures and Sums in Polygons 6.1 Properties 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. (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.

Example 3C: Finding Interior Angle Measures and Sums in Polygons 6.1 Properties of Polygons 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.

6.1 Properties of Polygons Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

6.1 Properties of Polygons 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.

6.1 Properties of Polygons 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.

6.1 Properties 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°.

6.1 Properties of Polygons An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Remember!

6.1 Properties of Polygons

Example 4A: Finding Interior Angle Measures and Sums in Polygons 6.1 Properties 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°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. measure of one ext.  = The measure of each exterior angle of a regular 20-gon is 18°.

Example 4B: Finding Interior Angle Measures and Sums in Polygons 6.1 Properties of Polygons Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

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

6.1 Properties 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 = 360 Combine like terms. r = 15 Divide both sides by 24.

6.1 Properties of Polygons Check It Out! Example 5 What if…? Suppose the shutter were formed by 8 blades instead of 10 blades. What would the measure of each exterior angle be? CBD is an exterior angle of a regular octagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular octagon has 8  ext. , so divide the sum by 8.

6.1 Properties of Polygons Lesson Quiz Find the value of x in the diagram. 2. Find the value of x in the regular heptagon. X = 30 51.4° 4. Find the measure of each exterior angle of a regular 15-gon. 24°