6-1 Properties and Attributes of Polygons Holt McDougal Geometry Holt Geometry
Warm Up 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
Each segment that forms a polygon is a side of the polygon 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.
You can name a polygon by the number of its sides You can name a polygon by the number of its sides. The table shows the names of some common polygons.
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Remember!
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
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
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
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
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
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
All the sides are congruent in an equilateral polygon 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.
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.
Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex
Example 2B: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave
Example 2C: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex
Check It Out! Example 2a Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex
Check It Out! Example 2b Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave
By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember!
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 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 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 Find the measure of each interior angle of pentagon ABCDE. Polygon Sum Thm. (5 – 2)180° = 540° Polygon Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.
Example 3C Continued mA = 35(4°) =140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°
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.
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.
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°.
An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Remember!
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 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.
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°.
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.
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.