Obj. 25 Properties of Polygons

• 1. Obj. 25 Properties of Polygons The student is able to (I can): Name polygons based on their number of sides Classify polygons based on concave or convex equilateral, equiangular, regular Calculate and use the measures of interior and exterior angles of polygons

2. polygonA closed plane figure formed by three or more noncollinear straight lines that intersect only at their endpoints.polygonsnot polygons 3. vertexThe common endpoint of two sides. Plural: vertices vertices.diagonalA segment that connects any two nonconsecutive vertices. diagonalregularvertexA polygon that is both equilateral and equiangular. 4. Polygons are named by the number of their sides: SidesName3Triangle4Quadrilateral5Pentagon6Hexagon7Heptagon8Octagon9Nonagon10Decagon12Dodecagonnn-gon 5. ExamplesIdentify the general name of each polygon: 1. pentagon2. dodecagon3. quadrilateral 6. concaveA diagonal of the polygon contains points outside the polygon. (caved in)convexNot concave.concave pentagonconvex quadrilateral 7. We know that the angles of a triangle add up to 180, but what about other polygons? Draw a convex polygon of at least 4 sides: 180 180 180Now, draw all possible diagonals from one vertex. How many triangles are there? What is the sum of their angles? 8. Thm 6-1-1Polygon Angle Sum Theorem The sum of the interior angles of a convex polygon with n sides is (n 2)180 If the polygon is equiangular, then the measure of one angle is (n 2)180n 9. SidesNameTrianglesSum Int.Each Int. (Regular)3Triangle1(1)180=180604Quadrilateral2(2)180=360905Pentagon3(3)180=5401086Hexagon7Heptagon8Octagon9Nonagon10Decagon12Dodecagonnn-gon 10. Lets update our table:SidesNameTrianglesSum Int.Each Int. (Regular)3Triangle1(1)180=180604Quadrilateral2(2)180=360905Pentagon3(3)180=5401086Hexagon4(4)180=7201207Heptagon5(5)180=900128.68Octagon6(6)180=10801359Nonagon7(7)180=126014010Decagon8(8)180=144014412Dodecagon10(10)180=1800150nn-gonn2(n 2)180(n 2)180n 11. An exterior angle is an angle created by extending the side of a polygon: Exterior angleNow, consider the exterior angles of a regular pentagon: 12. From our table, we know that each interior angles is 108. This means that each exterior angle is 180 108 = 72. 72 72 72 108 72 72The sum of the exterior angles is therefore 5(72) = 360. It turns out this is true for any convex polygon, regular or not. 13. Polygon Exterior Angle Sum Theorem The sum of the exterior angles of a convex polygon is 360. For any equiangular convex polygon with n sides, each exterior angle is 360n SidesNameSum Ext.Each Ext.3Triangle3601204Quadrilateral360905Pentagon360726Hexagon360608Octagon36045nn-gon360360/n