TEKS: G2B, G3B, G4A, G5A, G5B, G9B The student will determine the validity of conjectures. The student will construct and justify statements.

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2 TEKS: G2B, G3B, G4A, G5A, G5B, G9B The student will determine the validity of conjectures. The student will construct and justify statements. The student will select an appropriate representation to solve problems. The student will develop algebraic expressions representing geometric properties. The student will use patters to make generalizations about angle relationships in polygons. The student will formulate and test conjectures about the properties and attributes of polygons and their parts based on explorations.

3 In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments that intersect only at their endpoints.

4 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. This polygon is ABCDE or AEDCB or many other options. You may start at any letter and go in a circular motion either clockwise or counter-clockwise.

5 You can name a polygon by the number of its sides. The table shows the names of some common polygons.

6 Example: 1 Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon polygon, heptagon not a polygon not a polygon polygon, nonagon not a polygon

7 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.

8 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.

9 Example: 2 Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex irregular, concave regular, convex regular, convex irregular, concave

10 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 Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180. (1) 180 = 180 (2) 180 =360 (3) 180 =540 (4) 180 =720 (n-2) (n-2) 180

12 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.

13 Example: 3 Find the sum of the interior angle measures of a convex heptagon. (n 2)180 Polygon Sum Thm. (7 2) A heptagon has 7 sides, so substitute 7 for n. Simplify.

14 Example: 4 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 (16 2)180 = 2520 Polygon Sum Thm. Substitute 16 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are, so divide by 16.

15 Example: 5 Find the sum of the interior angle measures of a convex 15-gon. (n 2)180 Polygon Sum Thm. (15 2) A 15-gon has 15 sides, so substitute 15 for n. Simplify.

16 Example: 6 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 = c = 540 Substitute. Combine like terms. c = 4 Divide both sides by 135. m A = 35(4 ) = 140 m B = m E = 18(4 ) = 72 m C = m D = 32(4 ) = 128

17 Example: 7 Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n 2)180 (10 2)180 = 1440 Step 2 Find the measure of one interior angle. Polygon Sum Thm. Substitute 10 for n and simplify. The int. s are, so divide by 10.

18 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.

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

20 Example: 8 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. measure of one ext. = Polygon Sum Thm. 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.

21 Example: 9 Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360. measure of one ext. Polygon Sum Thm. 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.

22 Example: 10 Find the value of b in polygon FGHJKL. 15b + 18b + 33b + 16b + 10b + 28b = 360 Polygon Ext. Sum Thm. 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

23 Example: 11 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.

24 Example: 12 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.

25 Example: 13 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.