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