Presentation on theme: "3.5 The Polygon Angle-Sum Theorems"— Presentation transcript:
1 3.5 The Polygon Angle-Sum Theorems GeometryMr. Barnes
2 Objectives: To Classify Polygons To find the sums of the measures of the interior and exterior angles of polygons.
3 Definitions:SIDEPolygon—a plane figure that meets the following conditions:It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear.Each side intersects exactly two other sides, one at each endpoint.Vertex – each endpoint of a side. Plural is vertices.You can name a polygon by listing its vertices consecutively.For instance, PQRST and QPTSR are two correct names for the polygon above.
4 Example 1: Identifying Polygons State whether the figure is a polygon.If it is not, explain why.Not D- because D has a side that isn’t a segment – it’s an arc.Not E- because two of the sides intersect only one other side.Not F- because some of its sides intersect more than two sides.Figures A, B, and C are polygons.
5 Polygons are named by the number of sides they have – MEMORIZE Type of Polygon3Triangle4Quadrilateral5Pentagon6Hexagon7Heptagon
6 Polygons are named by the number of sides they have – MEMORIZE Type of Polygon8Octagon9Nonagon10Decagon12Dodecagonnn-gon
7 Convex or concave?Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.See how it doesn’t go on theInside-- convexSee how this crossesa point on the inside?Concave.
8 Convex or concave? CONCAVE CONVEX Identify the polygon and state whether it is convex or concave.CONCAVEA polygon is EQUILATERALIf all of its sides are congruent.A polygon is EQUIANGULARif all of its interior angles are congruent.A polygon is REGULAR if it isequilateral and equiangular.CONVEX
9 Ex. : Interior Angles of a Quadrilateral 80°Ex. : Interior Angles of a Quadrilateral70°2x°x°x°+ 2x° + 70° + 80° = 360°3x = 3603x = 210x = 70Sum of the measures of int. s ofA quadrilateral is 360°Combine like termsSubtract 150 from each side.Divide each side by 3.Find m Q and mR.mQ = x° = 70°mR = 2x°= 140°►So, mQ = 70° and mR = 140°
10 Investigation Activity Sketch polygons with 4, 5, 6, 7, and 8 sidesDivide Each Polygon into triangles by drawing all diagonals that are possible from one vertexMultiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon.Look for a pattern. Describe any that you have found.Write a rule for the sum of the measures of the angles of an n-gon
11 Polygon Angle-Sum Theorem The sum of the measures of the angles of an n-gon is(n-2)180Ex: Find the sum of the measures of the angles of a 15-gonSum = (n-2)180= (15-2)180= (13)180= 2340
12 ExampleThe sum of the interior angles of a polygon is How many sides does the polygon have?Sum = (n-2)1809180 = (n-2)18051 = n-253 = nThe polygon has 53 sides.
13 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.An equilateral polygon has all sides congruentAn equiangular polygon has all angles congruentA regular polygon is both equilateral and equiangular.