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Section 6-1 Properties of Polygons Classifying Polygons Polygon: Closed plane figure with at least three sides that are segments intersecting only at their endpoints, and no adjacent sides are collinear. B B C A A B C C A E D Not a polygon D E Not a polygon D E Polygon Classifying Polygons Note: A diagonal of a polygon is a segments that connects two nonconsecutive vertices. Convex Polygon: A polygon with no diagonal with points outside the polygon Concave Polygon: A polygon with at least one diagonal with points outside the polygon. Convex Concave Classifying Polygons – Example 1 Classify each polygon by its sides. Identify each as convex or concave. Convex Octogon Concave Hexagon Concave 20-gon Classifying Polygons Equilateral Polygon: A polygon with all sides congruent. Equiangular Polygon: A polygon with all angles congruent. Regular Polygon: A polygon that is both equilateral and equiangular. Classifying Polygons We can classify polygons according to the number of sides it has. Sides Name 3 Triangle Quadrilateral Pentagon Hexagon Octagon Nonagon Decagon Dodecagon n-gon 4 5 6 8 9 10 12 n Classifying Polygons – Example 2 Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. Regular, Convex Irregular; Concave Regular, Convex Naming Polygons D C G H F K J •To name a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction. Polygon: DGHJKFC or GDCFKJH Vertices: C,D,G,H,J,K,F Sides: Angles: DG, GH, HJ, JK, KF, FC, CD <D, <G, <H, <J, <K, <F, <C Class Work… Complete the table with a partner/ group Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the measures of the angles in an n-gon is (n – 2)180. Assignment #39 Pg. 386 #2-8 all AND DEFINE TERMS: Polygon, Convex Polygon, Concave Polygon, Equilateral Polygon, Equiangular Polygon, Regular Polygon, & Polygon Angle Sum Theorem Section 6-1 (cont.) Properties of Polygons Polygon Angle Sum Theorem Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the measures of the angles in an n-gon is (n – 2)180. Polygon Angle Sum – Example 3 Find the sum of the measures of the angles of a regular dodecagon. Then find the measure of each interior angle. = (n – 2)180 Polygon Angle Sum Theorem = (12 – 2)180 Substitution = (10)180 Simplify = 1800 Simplify **Be Careful!! Are they asking for the SUM of the angles, or for EACH angle measure?** Example 4 The sum of the angles of a polygon is 720. Find the number of sides the polygon has and classify it. Sum of the Angles = (n – 2)180 720 = (n – 2)180 4 = (n – 2) 6=n Hexagon Polygon Angle Sum Theorem Substitution Divide both sides by 180 Add 2 to both sides Example 5 Find x in the following polygon. 125° x° 125° To solve, we will use the polygon angle sum theorem for n=5 90 + 90 + 125 + x + 125 = (5 – 2)180 Polygon Angle Sum Theorem 330 + x = (5 - 2)180 330 + x = 540 x = 210 Simplify Simplify Subtract 330 from both sides Example 6 Find the measure of each angle in the following polygon. To solve, we will use the polygon angle sum theorem for n=5 (5 x 2) (5 x 10) (4 x 15) (8 x 8) (3x 5) (5 2)(180) Polygon Angle Sum Theorem Simplify / Combine Like Terms 25 x 40 (3)(180) Simplify 25x 40 540 Solve for x 25x 500 x 20 Now, solve for each angle in the polygon… A 5x 2 5(20) 2 102 E 5x 10 5(20) 10 120 D 4x 15 4(20) 15 95 C 8x 8 8(20) 8 168 65 B 3x 5 3(20) 5 Polygon Angle Sums Theorem 3-15: Polygon Exterior Angle-Sum Theorem: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. For a pentagon: m1 m2 m3 m4 m5 360 3 2 4 1 5 Polygon Angle Sums – Example 7 Find the measure of each exterior angle of a regular dodecagon. **The sum of the exterior angles is 360 degrees** Set up an equation **Remember, a regular polygon has both equal sides and equal angles Exterior Angle Sum= Measure of Each Angle (Number of Sides) 360 = x(12) Now, Solve. 30 = x Divide both sides by 12 Each exterior angle is 30 degrees Example 8 Find the measure of each angle in the following polygon. To solve, we will use the polygon exterior angle sum theorem & set all angles equal to 360 33b 18b 15b 28b 10b 16b 360 Polygon Exterior Angle Sum Theorem Simplify / Combine Like Terms 120b 360 Solve for b b3 Now, solve for each angle in the polygon… H 33b 33(3) 99 G 18b 18(3) 54 F 15b 15(3) 45 L 28b 28(3) 84 K 10b 10(3) 30 J 16b 16(3) 48 Assignment #40 Pg. 386 #9-15 all #16-21 all